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MUSICOLOGY 

A  TEXT-BOOK  FOR  SCHOOLS  AND  FOR 
-:-   -:-   -:-   GENERAL  USE   -:-   -:-   -:- 


By 

MAURICE  S.   LOGAN 


HINDS.  NOBLE  &  ELDREDGE.   Publishers. 

31.  33.  35  West  15  th  Street. 

New   York   City. 


Copyright.   1909. 

BV 

MAURICE    S.    LOGAN. 


r:^  '^     c/^^ 


Entered  at  Stationer's  Hall,   London.   Ent^land. 


Printed  in  the  United  States  of  America. 


PREFACE 


The  object  of  this  work  is  to  furnish  a  practical  and  com- 
prehensive text-book  on  the  theory  and  philosophy  of  music, 
for  schools  and  for  general  use.  It  deals  with  the  science  rather 
than  the  art,  while  most  music  text-books  deal  with  the  art 
rather  than  the  science. 

For  school  use  it  is  intended  to  be  included  in  the  science 

course,  rather  than  in  the  music  course.     From  the  educational 

standpoint,  either  of  mental  training  or  useful  knowledge,  the 

science  of  music,  as   distinct  from  the  art,  is  entitled  to  rank 

with  the  other  sciences  taught. 

M.  S.  L. 


iv;58?947 


TABLE  OF  CONTENTS 


PART  FIRST. 
ELEMENTARY. 


PAGE 

Common  Terms  and  Signs  Used  in  Music      -  -  -  -      17 

Notes,  1—  Rests,  2—  Staff,  4—  Clefs,  7—  Score,  14—  Sharps, 
Flats,  and  Naturals,  20 —  Signature,  21 —  Accidentals,  22. 

Rhythm  -  -  ------      22 

Accent,  1 —  Bars,  3 —  Time,  5 —  Duple.  Triple,  6 —  Simple, 
Compound,  12 —  Double,  Triple,  Quadruple,  12 —  Broken 
Pulses,  14 —  Tataing,  16 —  Counting  Time,  22 —  Measures, 
3  and  23 —  Tempo  or  Rate  of  Movement,  28 —  Standard  of 
time  value,  29 —  Applying  Words  to  Music,  30 —  Interrupted 
Rhythms,  34 —  Change  of  Tempo,  Pause,  Syncopation,  34 — 
Triplets,  36 —  Repeats^  37. 

Expression  -  -  ------29 

Terms  Relating  to  Power,  2 —  Terms  Relating  to  Quality, 
3 —  Terms  Relating  to  Movement,  4 —  Embellishments,  6 — 
Turn,  6 —  Trill,  7 —  Mordent,  8 —  Grace  Notes,  12 —  Appog- 
giaturas,  13 —  Acciaccaturas.  14 —  After  Tones,  15. 

Keys  ---------      31 

Chart  I,  2—  Key  Patterns,  4—  Major,  5—  Minor,  6—  Forma- 
tion of  Signatures,  15 —  Intermediate  Tones,  17 —  Keyboard, 
18—  Chart  II,  19—  Diatonic  Scale,  22—  Chromatic  Scale,  23. 

Re.\ding  Music  -  -  -  --  -  -35 

Running  the  Scale,  1 —  Mental  effect  of  each  Key  tone,  2 — 
Transposition,  5 —  Key-Tonality,  7 —  Hearing  with  the  eye, 
8 —  Seeing  with  the   ear,  8 —  Solfaing,  12 —  Laing,  13. 

Tonic  Solfa  Notation  -  -----      37 

Interrelationship  of  Major  Keys  -  -  -  -      38 

Key  Building,  1 —  Major  Key-Table,  5 —  Analysis  of  Table,  6 — 
Locating  Key-note,  11 —  Enharmonic  Change,  13 —  Key  Circle, 
15 —  Relation  of  Sharp  and  Flat  Keys  on  same  letter,  16. 

Interrelationship  of  Major  and  Minor  Keys  -  -  -      43 

Relative  Minor,  1 —  Tonic  Minor,  6 —  Combined  Major  and 
Minor  Key-Table,  12 —  Analysis  of  '  Table  from  relative  and 
tonic  views,  16 —  Double  sharps  and  flats,  41 

Melodic  Minor  Scale  -  -  -  -  -  -      52 

Old  Minor  Mode  -  -  -  -  -  -  -53 


12  TABLE  OF  CONTEXTS 

PAGE 

Ancient  Greek  Modes  -  -  -  -  -  -      54 

Distinction  between  modes  and  keys,  3. 

Harmonic  Scale  Names  -  -  -  -  -  -      55 

Dominant  and  Sub-Dominant  sides  of  the  scale,  1. 

Intervals  -  -  ------      55 

Standards  of  measurement,  2 — Major  and  perfect  intervals,  2 — 
Analysis  of  the  scale,  2 —  Simple  and  compound  intervals,  2 — 
Minor,  diminished,  and  augmented  intervals,  3 —  Inverted 
intervals,  6 —  Diatonic  and  Chromatic  intervals.  7 —  Concordant 
and  discordant  intervals,  12 —  Formation  of  the  Diatonic  and 
Chromatic  Scales,  14. 


PART  SECOND. 
STRUCTURE  OF  MUSIC. 

Harmony  -  -  -  -      59 

Chords  and  Discords,  1 —  Building  in  3rds,  2 —  Harmonic 
Scale,  3 —  The  Triad.  4 —  Four  Part  Harmony,  7 —  Major, 
Minor,  Dim.  and  Aug.  Triads,  11,  12 —  Analysis  of  the  triads  of 
the  major  and  minor  modes,  13,  14 —  Characteristic  Harmonies, 
19—   Laws.   21.    23. 

Positions  and  Forms  of  Triads  -  -  -  -  -      64 

Thorough   Bass  Figuring  of  Triads     '  -  -  -  -      66 

Dissonant  Chords  -  -  -  -  -  -  -67 

Dissonant  Triads,  3 —  Extensions  of  the  Triad,  6 —  Chords 
of  the  7th,  11 —  Classification  and  analysis  of  chords  of 
the  7th.  11 —  Positions  and  Forms  of  Chords  of  the  7th, 
17 —  Thorough  Bass  Figuring  of  Chords  of  the  7th,  18 — 
Chord  of  the  9th,  20 —  Chords  of  the  6th,  23. 

Interrelationshii'  of  Chords   in  General      -  -  -  -      74 

Common  Key  Relationship,  2 —  Root  Relationship,  r, —  Com- 
mon Tone  Rel.\tionship.  6. 

Interrelationship  of  the   Triads    of   a    Key  -  -  -      75 

Progression  --------76 

General  principles,  1 —  Progression  of  Triads,  6 —  Voice 
Leading,  6 —  Skips,  14 —  Progression  of  the  Bass,  17 —  Con- 
secutive .")ths  or  Octaves,  19 —  Hidden  Consecutives,  24 —  False 
Relations,  26 —  Motion  of  the  Parts,  28 —  Preparation  of 
Dissonances.  29. 

Resolution  .  .  -  .  -  .  _  _      g2 

General  principles,  1 —  Tendency  of  Dissonances  to  resolve,  6 — 
Rules  of  Resolution.  8 —  Ornamental  Resolution.  12 —  Resolu- 
tion of  Dissonant  Chords,  14 —  Treatment  in  upper  part.  19 — 
Treatment  in  lower  part,  19 —  Formulas  for  the  Resolution  of 
7th  Chords,  19—  When  the  rules  of  Resolution  are  set  aside, 
22. 


TABLE    OF    CONTENTS  13 

PAGE 

Suspension  and   Anticipation  -  -  -  -  -      87 

Object,  3—  General  Rule,  5—  Principal  Suspensions,  6. 

Passing  Notes  ..-----87 

Pedal   Passage  -  -  -  -  -  -  -83 

The   Cadence  -------89 

Authentic:  Plagal,  1—  Half  Cadence,  2—  Deceptive  Cadence, 
4 —  Cadenza,  5. 

Sequence  --------90 

Modulation  .-------90 

General   principles,  1 — 11. 

Methods  of  Modulation  -  -  -  -  -  -      93 

Modulation  by  Means  of  Connecting  Triads,  2—  Table  of 
the  triads  common  to  different  keys,  3 —  Common  triad  form- 
ulas, 7 —  Confirming  the  Modulation,  12—  Modulation  by 
Means  of  Connecting  Tones,  15 —  General  principles,  15 — 
Major,  minor,  and  dim.  triad  formulas  combined,  20 —  Applica- 
tion of  the  Combination  Formula,  21 —  Modulation  with  the 
Dominant  7th  Chord,  24 —  The  modulating  properties  of  the 
Dom.  7th  Chord,  25 —  The  modulations  resulting  from  forming 
the  Dom.  7th  Chord  on  the  different  tones  of  a  key,  28 —  When 
the  modulation  leads  to  major,  and  when  to  minor  keys,  42 — 
Resolving  the  Dom.  7th  Chord  of  one  key  to  the  Dom.  7th 
Chord  of  any  other  key,  4.3 —  Modulation  by  chromatic  runs, 
44 —  Modulation  with  the  Diminished  7th  Chord,  46 — 
The  modulating  properties  of  the  Dim.  7th  Chord,  47 —  Enhar- 
monic modulation  through  the  Dim.  7th  Chord,  54 —  The 
modulating  capacity  of  the  Dim.  7th  Chord,  60 —  Practical  limit 
of  modulation  in  either  direction,  62 —  Modulation  with  the 
Augmented  6th  Chord,  63 —  Italian,  French  and  German  6th 
Chords,  64 —  Their  common  resolution,  68 —  The  natural  seat 
of  the  Aug.  6th  Chord,  70 —  Their  modulating  tendency,  71 — 
Modulation  by  enharmonically  changing  the  German  6th  Chord 
into  a  Dom.  7th  Chord,  and  vice  versa,  72 —  Modulation  by 
Inversion  (over  to  page  120) —  Modulation  by  Imitation 
(over  to  page  125). 

Summary  of  Modul.\tion  ------    107 

Diatonic  and  chromatic  tones,  1,  2 —  Diatonic  and  chromatic 
chords,  3 —  Modulation  without  or  in  advance  of  accidentals, 
5 —  Three  stages  of  modulation.  7 —  Classification  of  Modula- 
tion, 9 —   When  to  modulate,   14. 

Transposition         -         -  -  -  -  -  -  -    109 

Distinction  between  Modulation  and  Transposition,  1 — 
Transposition  merely  a  change  of  pitch,  2 —  Singing  in  a  dif- 
ferent- key  from  that  in  which  the  music  is  written,  7 —  Play- 
ing in  a  different  key  from  that  in  which  the  music  is  written,  8. 


14  TABLE  OF  CONTENTS 

PAGE 

Counterpoint  -  -  -    112 

Distinction  between  Harmony  and  Counterpoint  2 —  Origin 
of  Counterpoint,  4 —  Outline,  7 —  Simple  Counterpoint,  8 — 
Contrapuntal  Rules,  12 —  Double  Counterpoint,  28 —  Table 
of  Inversions,  29 —  Analysis  of  Inversions,  31 —  Modulation 
BY  Inversion^  44 —  Counterpoint  Invertible  in  Various 
Intervals,  53 —  Triple  and  Quadruple  Counterpoint,  56. 

Imitation  --------    123 

Kinds  of  Imitation,   4 —  Modulation   by   Imitation,   13. 

Contrapuntal  Music  -  _  -  .  _  -    126 

Fugues,  2 —  Canons,  10. 

Melody  -  -  -  -    130 

Interrelationship  of  Melody,  Counterpoint,  and  Harmonj^  1 — 
Chords  and  Alelodic  Figures,  5 —  Harmonic,  melodic,  and 
rhythmic  elements  of  figures,  7,  11 —  Simple  and  compound 
figures,  11 —  Development,  12 —  Thematic  Treatment,  13. 

Linguistic  Character  of  Music  -  .  _  .  .    133 

The    Period  ----.-_.    134 

Rhythmic  structure,  3 —  Thematic  structure,  9. 

Form  -  -  -  -    137 

Strophe  Form,  5 —  Art  Song  Form,  6 —  Unity.  Contrast,  and 
Symmetry,  10 —  The  Song  Forms,  16 —  OutHne.  18 —  The 
Scherzo  Minuet  or  Applied  Song  Form,  19 —  Outline — 
Rondo  Form,  23 — Outline —  Sonata  Form,  25 —  Outline — 
Suite  Form,  28 —  Outline —  Contrapuntal  Forms  (back  to 
pp.  126-129) —  Introductory,  Intermedi.\te,  and  Concluding 
Parts,  35—  Outline, 

Classification  of  Music  -  -  -  -  -  -    144 


PART  THIRD. 

ACOUSTICS. 

Acoustics  -  -  -  -  -    147 

Cause  of  sound,  1 —  Range  of  sound,  2 —  Sound  waves,  4 — 
Laws  of  Vibr.vtion,  7 —  Compound  Tones,  16 —  Compound 
vibration  of  strings,  16 —  Harmonic  Series,  19 —  Simple  Tones, 
24 —  EfTecl  of  overtones,  26 —  Chord  Formation,  35. 

Scale   of   Nature  ----_..    157 

Interval  Ratios,  2 —  Standards  of  Pitch,  8 —  Application  of 
ratios,  13 —  The  Enharmonic  Scales,  16. 


TABLE    OF    CONTENTS  1 5 

PAGE 

Equal  Temperament       -  -  -  -  -  -  -    162 

The  Equal  Temperament  Scale,  1 —  Application  of  logarithms, 
9 —  True  and  tempered  intervals  compared :  1st,  through 
their  logarithmic  values,  18;  2d,  through  their  semitone  values, 
20. 

Consonance  -  -  -  -  -  -  -  -    167 

Blending  of  overtones,  1 —  Blending  of  vibrations,  10. 

Beats  -  .  -  .  -  ....    170 

Imperfect  Unison  Beats,  5 —  Imperfect  Consonance  Beats, 
9—  Beat  Tones,  16. 

Differential  and  Summational  Tones  .  _  -  -    174 

Orders  of  Differentials,  5 —  Complimental  relations,  8 — 
Theories,  16. 

Dissonance  -_--._--    179 

As  due  to  beats,  1 —  As  due  to  combinational  tones,  6, 
Sympathetic  Resonance  -  .  .  .  .  _    igo 


PART  FOURTH. 
PRINCIPAL  SOURCES  OF  MUSICAL  SOUND. 

Outline  -  .  _  .  .  -  -  -    185 

Strings,  Rods,  Pipes,  Reeds  -  -  .  .  .    ige 

Transverse  Vibrations  of  Rods,  3 —  Longitudinal  Vibra- 
tions OF  Rods,  7 — Pipes,  14 —  Resonance  Boxes  and  Sounding 
Boards,  27—  Reeds,  28. 

Membranes,  Plates.  Bells  --.-..     192 

Membranes  and  Plates,  2 —  Bells,  14. 


PART  FIFTH. 

APPENDIX. 

History  of  the  Diatonic  Scale  -  -  _  .  .    197 

Ancient  Greek  Modes  ---...    202 

The  Earlier  Greek  Modes,  4 —  The  Later  Greek  Modes,  7 — 

The  Early  Church  Modes.  11. 

Table  of  Common  Musical  Intervals  _  .  .  .  2O6 

Temperament  -  --__..  207 

Equal  Temperament,  3 —  Mean-Tone  Temperament,  5. 

Tuning  -  -.-_.._  2II 

Calculation  of  Pitch  Numbers  -  .  .  .  .  214 

Character  of  the  Different  Keys  _  .  _  _  217 


PART    FIRST 


ELEMENTARY 


COMMON   TERMS   AND  SIGNS   USED    IN   MUSIC 

1 .  Notes  are  characters  used  in  music  to  represent  tones. 
As  to  relative  time  value,  there  are  six  kinds  of  notes  in  com- 
mon use :  the  whole  note  («>  ),  the  half  note  (J),  the  quarter 
note  (J),  the  eighth  note  (^^),  the  sixteenth  note  (  1^),  and  the 

thirty-second  note  (  ,s ) :  each  being  half  the  time  value  of  the 
preceding  one.  The  stems  may  extend  either  up  or  down ; 
and  the  hooks,  either  right  or  left.  Two  or  more  notes  are 
frequently  attached  to  one  stem.  When  necessary,  the  direc- 
tion of  the  stem  is  made  to  show  the  part  to  which  the  note 
belongs.  When  a  note  has  two  stems  in  opposite  directions, 
it  belongs  to  two  parts. 

2.  Rests  are  characters  to  represent  silence.  There  are 
six  rests,  corresponding  in  time  value  to  the  six  notes:  the 
whole  rest  (— ■),  the  half  rest  (-"),  the  quarter  rest  {"<*,  I,  or  ^), 
the  eighth  rest  (•/),  the  sixteenth  rest  (^),  and  the  thirty-sec- 
ond rest  ( ^ ). 

3.  A  dot  after  a  note  or  rest  increases  its  time  value  one- 
half;  thus  a  J.  is  equal  to  jj  j,  a  J.  is  equal  to  ^^  J^  J^,  and  a 
^  .  is  equal  to  ^^^^^^ — and  similarly  as  to  rests. 

4.  A  Staff  consists  of  five  lines  with  its  four  intervening 
spaces,  on  which  the  notes  are  written.      It  is  a  kind  of  musi- 


i8 


MUSICOLOGY 


cal  ladder  to  indicate  the  pitch  of  tones.  When  necessary, 
short  hnes  are  added  above  or  below.  Each  line  or  space  of 
the  staff  is  called  a  Degree. 


Treble  Staff. 


Bass  Staff. 


For  Soprano  and 
Alto  voices. 


For  Bass  and  Tenor 
voices. 


Fig.  1. 

(Base  in  music  is  usually  spelled  dass  ;  but  dase  is  more  in  accord  with  its  meaning,  as  it 
means  the  base  or  foundation.) 

5.  In  music  the  treble  and  bass  staffs  are  usually  sepa- 
rated so  that  words  of  a  song  may  be  written  between,  but 
it  must  be  remembered  that  their  relation  to  each  other  (as 
shown  in  Fig.  i)  is  not  changed.  Therefore,  when  thus  sep- 
arated, the  second  added  line  above  the  bass  staff  is  the  same 
as  the  lower  line  of  the  treble  staff,  and  the  second  added  line 
below  the  treble  staff  is  the  same  as  the  upper  line  of  the 
bass  staff. 

6.  Sometimes  (in  choir  or  other  music)  each  voice,  or  part, 
is  written  on  a  separate  staff. 

7.  The    S  clef  is  always  placed   on  G,  above   middle    C 

(encircling  the  line  marked  G),  and  is  therefore  called  the  G 
clef  (or  Treble  clef).  When  this  clef  is  used,  middle  C  is 
always  on  the  first  added  line  below  the  staff. 

8.  The    ^  clef  is    always   placed   on  F,  below  middle   C 

(starting  on  and  encircling  the  line  marked  F),  and  is  therefore 
called  the  F  clef  (or  Bass  clef).  When  this  clef  is  used,  mid- 
dle C  is  always  on  the  first  added  line  above  the  staff.     It  will 


ELEMENTARY  I 9 

be  seen  in  Fig.  i  that  middle  C  is  exactly  midway  between 
the  G  and  F  clefs. 

9.  The  jSj:  clef,  when  used,  is  placed  on  middle  C  (center- 
ing on  the  line  or  space  marked  C),  and  is  therefore  called 
the  C  clef  (or  Tenor  clef   when   used   for  the   tenor;    or  Alto 

clef  when  used  for  the  alto).      The  C  clef|  M:  I  is   a   movable 

clef,  and  may  be  used  to  locate  middle  C  on  any  degree  of 
the  staff  by  centering  it  on  the  desired  line  or  space.  It  must 
be  fixed  in  the  mind  that  middle  C  is  the  same  tone  wherever 
written.  It  is  called  middle  C  because  it  is  the  middle  of  the 
great  vocal  compass,  and  is  therefore  the  center  around  which 
the  different  voices  range :  female  voices  ranging  mostly 
above ;  and  male  voices,  mostly  below.  (See  the  average 
range  of  the  different  voices  in  Fig.   i.) 

10.  The  clefs  are  used  for  the  purpose  of  adjusting  the 
staff  to  the  range  of  voices  that  are  intended  to  sing  from  it. 

It  will  be  seen  from  Fig.  i  that  the  G  I  ^  I  clef,  by  locating  G 

on  the  second  line,  adjusts  the  staff  to  the  range  of  soprano 
voices  ;    and  that  the  F  (  ^'  j  clef,  by  locating  F  on  the  fourth 

line,  adjusts  the  staff  to  the  range  of  bass  voices.  But  we 
see  that  the  alto  and  tenor  (especially  the  tenor)  ranges  ex- 
tend over  onto  both  staffs  thus  adjusted:  the  alto,  notes 
(written  on  the  treble  staff)  would  require  added  lines  be- 
low the  staff,  or  the  tenor  notes  (written  on  the  bass  staff) 
would  require  added  lines  above  the  staff  (supposing  the  staffs 
separated  as  we  find  them  in  music).      Foi    this  reason  the  C 

I  ^  I  clef  is  sometimes  used  to  adjust  the  staff  to  the  alto  or 
the  tenor  (especially  the  tenor)  range  by  locating  middle  C 
(which  is  near  the  middle  of  both  ranges)  on  any  desired  de- 
gree of  the  staff  (usualh'  on  the  third  space  or  fourth  line  for 
tenor  or  on  the  third  line  for  alto). 

11.  The   tenor    staff    (middle    C   on   third    space)   and   the 


20  MUSICOLOGV 

treble  staff  (upper  C  on  third  space)  are  both  lettered 
alike  and  therefore  read  alike.  For  this  reason  the  treble  or 
G  clef  is  sometimes  used  for  writing  the  tenor;  but  the  tenor 
is  thus  written  an  octave  higher  than  sung. 

12.  A  staff  consists  merely  of  five  lines  (and  the  spaces  be- 
tween); its  range  being  determined  by  the  clef  placed  upon 
it.  Observe  that  the  staff  is  adjusted  to  the  clef,  and  not  the 
clef  to  the  staff;  for  each  clef  represents  a  fixed  tone. 

13.  As  the  C  clef  represents  middle  C,  its  natural  position 
is  exactly  midway  between  the  G  and  F  clefs. 

14.  When  the  music  for  the  different  voices  or  instruments 
is  written  on  separate  staffs,  it  is  called  score  music,  and  two 
or  more  staffs  connected  by  a  brace  is  called  a  Score. 

15.  In  most  song-books,  however,  we  find  the  soprano 
and  alto  written  together  on  the  treble  staff,  and  the  bass 
and  tenor  together  on  the  bass  staff.  In  which  case  the  so- 
prano is  represented  by  the  upper  notes  on  the  treble  staff, 
and  the  alto  by  the  lo\\'er  notes  on  the  treble  staff;  the  tenor 
is  represented  by  the  upper  notes  on  the  bass  staff,  and  the 
bass  by  the  lower  notes  on  the  bass  staff. 

16.  The  tune,  or  melody,  is  usually  in  tlie  soprano — the 
other  parts  being  an  accompaniment. 

17.  Female  voices  range  one  octave  (eight  degrees)  higher 
than  male  voices — female  voices  being  limited  to  the  soprano 
and  alto  ranges,  and  male  voices  to  the  bass  and  tenor  ranges 
— so  that  men  in  singing  the  soprano  or  alto  from  the  treble 
staff  sing  an  octave  lower  than  written,  and  women  in  sing- 
ing the  bass  or  tenor  from  tlie  bass  staff  sing  an  octa\-e  higher 
than  written.  This,  howexer,  occasions  no  incoiuenience,  as 
singing  in  octaves  produces  the  same  musical  effect ;  for 
which  reason,  tones  an  octave  apart  are  lettered  alike. 

18.  The  bass  and  treble  staffs  (shown  in  Fig.  i)  are  fixed 
as  regards  their  tone  range,  so  that  each  line  and  each  space 
represents  a  certain  pitch  of  tone,  each  of  which  is  named  by 
one    of   the   seven   letters.    A.  H.  C.  1).  E,  I*",  G  ;    these  seven 


ELEMENTARY 


21 


letters  being  repeated   in  each   octave.      (I'rom   any  letter  to 
the  same  letter  above  or  below  is  an  octave.) 

19.  The  names  usually  applied  to  the  relative  tones  of  a 
key  (see  p.  31:1)  are  the  seven  syllables  Do,  Ri\  J//,  Fa,  Sol, 
La,  Ti,  which  are  repeated  in  each  octave.  These  syllables 
are  a  great  assistance  in  reading  music,  and  aid  in  producing 
good  tones.  They  are  of  Italian  origin,  and  must  be  given 
the  Italian  pronunciation  by  giving  i  the  sound  of  e,  e  the 
sound  of  a,  a  the  sound  of  a  in  "far,"  and  o  the  sound  of  o  in 
"no."  For  the  sake  of  uniformity  in  the  different  coun- 
tries, the  Italian  language  is  recognized  as  the  language  of 
music. 

20.  A  Sharp  (ji)  placed  on  a  line  or  space  raises  its  pitch  a 
half-step.  A  Flat  ( b )  placed  on  aline  or  space  lowers  its 
pitch  a  half-step.  A  Natural  (^)  destroys  the  effect  of  a 
sharp  or  a  flat. 

21.  The  sharps  or  flats  at  the  beginning  of  a  piece  of  music 
is  called  the  Signature.  They  are  placed  on  the  lines  and 
spaces  that  are  to  be  sharped  or  flatted  through- 
out the  piece.  They  also  affect  the  same  letters 
in  each  octave. 

22.  Sharps, flats,  and  naturals  are  called  Accid- 
entals when  they  occur  during  the  course  of 
the  music.  An  accidental  influences  the  line 
or  space  on  which  it  is  placed  only  to  the  end 
of  the  measure,  unless  the  affected  note  is  tied 
over  into  the  next  measure. 

23.  In  reading  music  thus  affected  by  acci- 
dentals, to  give  a  sharp  effect  to  any  key  syllable 
use  the  sound  of  i  (pronounced  c) ;  thus  chang- 
ing Do  to  Di,  Re  to  Ri,  Fa  to /^/,  Sol  to  Si,  and 
La  to  Li.  To  give  a  flat  effect  to  any  key  syllable  use  the 
sound  of  r  (pronounced  rt) — in  Re  use  the  sound  of  ^  (pro- 
nounced a) — thus  changing  Ti  to  Tc,  La  to  Lc,  Solt  o  Se,  Mi 
to  JSIc,  and  Re  to  Ra. 


b 

~Do- 
-Ti~ 

« 

Te 

-La- 

Li 

Le 

Sol- 

Si 

Se 

-Fa- 
-Mi- 

Fi 

Me 

-Re- 

Ri 

Ra 

- 

Di 

-Do-^ 

Give    Italian    pro 

nur.ciation. 

F 

IG.  3. 

22  MUSICOLOGV 

RHYTHM 

1 .  Accent  is  a  stress  of  the  voice  on  certain  pulsations  of 
the  music. 

2.  Music  has  regularly  recurring  accents  which  cause  it  to 
throb  or  pulsate.  This  throb  or  pulsation  is  called  the 
Rhythm  of  the  music. 

3.  A  light  bar  is  drawn  across  the  staff  before  each  heavy 
accent,  thus  dividing  the  music  into  equal  parts  called  Meas- 
ures.     Measures  therefore  indicate  the  rhythm  of  the  music. 

4.  Heavy  bars  across  the  staff  are  frequently  used  to  mark 
the  end  of  a  line  of  words.  A  double  bar  across  the  end  of 
the  staff  marks  the  close  of  the  music  or  of  a  strain. 

5.  Time.  Time  refers  to  the  number  and  time  value  of 
the  pulses  in  each  measure,  by  which  the  time  value  as  well 
as  the  rhythmic  nature  of  each  measure  is  shown. 

6.  The  time  value  to  be  given  to  each  measure  is  shown 
by  the  figures  at  the  beginning  of  the  music,  called  the  Frac- 
tion; the  upper  figure  showing  the  number  of  pulses  in  each 
measure,  and  the  lower  figure  showing  the  time  value  of  each 
pulse.  (|  time  is  sometimes  marked  thus,  g;  |  time  thus, 
g;  and  \  time  thus,  ^  ^  or  g  g.)  The  lower  figure  of  the 
fraction  is  usually  2,  4,  or  8,  according  as  each  pulse  is  to  be 
valued  at  a  half,  quarter,  or  eighth  note.  The  upper  figure 
is  usually  2,  3,  4,  6,  9,  or  12.  If  the  upper  figure  is  divisible 
by  2,  it  is  called  duple  or  even  time  (or  rhythm);  if  divisible 
by  3,  it  is  called  triple  or  uneven  time  (or  rhythm). 

7.  In  duj)le  time  every  other  pulse  is  accented.  In  triple 
time  every  third  pulse  is  accented.  6  and  12  naturally  divide 
into  groups  of  3's  by  accents  on  every  third  pulse,  and  are 
therefore  triple  as  to  the  groups,  though  even  as  to  number 
of  groups,  in  a  measure.  9  is  triple,  both  as  to  groups  and 
number  of  groups. 

8.  The  heavy  accent  in  each  measure  is  always  on  the  first 
note  of  the  measure;  the  other  accents  (if  any)  in  each  meas- 
ure are  lighter  accents.      As  every  second  or  third  pulse  is  ac- 


ELEMENTARY  23 

cented,  therefore,  if  a  measure  contains  four  or  more  pulses, 
it  must  contain  one  or  more  light  accents;  and  the  pulses 
will  be  divided  by  means  of  the  accents  into  groups  of  2's  or 
3's,  according  as  the  time  is  duple  or  triple. 

9.  Sometimes  the  upper  figure  of  the  fraction  is  5  or  7. 
These  are  called  peculiar  rhythms,  being  neither  duple  nor 
triple,  but  may  divide  into  alternate  groups  of  2's  and  3's, 
and  therefore  alternately  duple  and  triple  as  to  the  groups. 

10.  Two  dissimilar  rhythms  are  sometimes  used  at  the 
same  time  (but  in  different  parts)  and  called  combined 
rhythm. 

11.  The  usual  method  of  keeping  time  in  singing  is  by 
beating  time  with  the  hand.  The  usual  manner  of  beating 
time  is — down,  up;  down,  left,  up;  or  down,  left,  right,  up 
— according  as  there  are  2,  3,  or  4  pulses  to  the  measure,  in- 
cluding one  pulse  to  each  beat.  If  there  are  6,  9,  or  12 
pulses  in  a  measure  (divided  into  groups  of  3's),  the  time  is 
beat  in  the  same  manner  by  including  three  pulses  to  each 
beat. 

12.  If  one  pulse  is  included  in  each  beat,  the  time  is  called 
Simple.  If  three  pulses  are  included  in  each  beat,  the  time 
is  called  Compound.  As  regards  the  number  of  beats  in  a 
measure,  the  time  is  commonly  called  double,  triple,  or  quad- 
ruple (if  simple) ;  or,  compound  double,  triple,  or  quadruple 
(if  compound).      In  compound  time,  only  the  accents  are  beat. 

13.  The  manner  of  beating  time  is  not  essential- — evenness 
being  the  essential  feature.  Beating  time  is  not  difTficult  if 
the  rhythm  or  throb  of  the  music  is  felt :  feeling  the  rhythm 
is  beating  time  mentally.  A  common  method  of  keeping 
time  when  playing  is  counting  the  pulses  in  each  measure. 

14.  Broken  Pulses.  If  the  time  value  of  a  pulse  is  broken 
in  any  manner,  it  is  called  7k  broken  pulse.  The  time  value  of 
a  pulse  maybe  broken  up  into  any  kind  of  notes,  or  notes  and 
rests,  which  together  equal  the  time  value  of  the  pulse.  On 
the  other  hand,  a  single  note  may  cover  the  time  value  of  one 


24 


MUSICUI.UGV 


pulse  and  part  of  anotlier  or  two  or  more  pulses.  Thus  there 
is  no  fixed  relation  between  the  time  values  of  the  pulses  and 
the  notes.  If  their  time  values  corresponded  each  to  each, 
keeping  time  in  music  would  be  a  very  simple  matter.  This 
is  the  case  in  some  very  simple  pieces  of  music.  The  diflfi- 
culty  in  keeping  time  correctly  naturally  increases  as  the  rela- 
tion between  the  time  values  of  the  pulses  and  the  notes 
becomes  more  and  more  complex. 

15.  Beating  time  serves  to  mark  the  time  value  of  the 
pulses  (singly,  in  simple  time;  by  3's  or  by  accents,  in 
compound  time),  so  that  the  real  art  of  keeping  time  consists 
in  giving  the  proper  relative  time  values  to  the  different  parts 
of  the  broken  pulses.  This  involves  a  development  of  the 
sense  of  relative  time.  This  sense  is  naturally  stronger  in 
some  persons  than  in  others,  but  it  is  usually  more  or  less  a 
matter  of  development. 

16.  Tataing  is  a  method  which  has  been  found  useful  in 
developing  the  sense  of  relative  time.  It  consists  in  giving 
time  names  to  the  pulse  and  its  various  divisions,  which  must 
become  associated  with  time  just  as  the  key  syllables  are 
associated  with  pitch, 

17.  The  following  is  a  diagram  of  a  system   of  Tataing: 


a 

h 

c 

d 

e 

f 

9 

1 

1     1 

1 

1 

Ti" 

1 
"3 

1111 

i     4      4      4 

1 

1       1 

4        4 

1       1 
4       4 

I 

3  1 

4  4 

4-       ^ 

T           4 

Tii 

lii-la 

tii 

lii  - 

tc 

ta-fa-tc-fc 

tii 

t(5  -  fe 

ta  -  fa 

15 

tii  -  e  -  fe 

tii  -  fa  -  a 

(.Pronounce  vowels  as  marked.) 
Fu;.  3. 

18.  Td  is  the  name  of  the  whole  pulse  regardless  of  the 
time  value  of  the  pulse  note.  If  it  extends  over  one  pulse, 
the  vowel  is  repeated  (Ta-a)  for  each  pulse  over  which  it  ex- 
tends. The  sections  n,  b,-  c,  d,  e,  f,  g  represent  broken 
pulses — each  broken  as  indicated  by  the  fractions.  The  value 
of  the   notes    corresponding  to   these   fractions    depends,    of 


ELEMENTARY  25 

course,  on  the  value  of  the  pulse  note.      The  fractions  |  rep- 
resent dottetl  notes. 

19.  Observe  that  d  is  made  up  of  the  first  half  of  a  and 
the  last  half  of  r ;  e  is  made  up  of  the  first  half  of  rand  the 
last  half  o{  a  \  /is  the  same  as  d  except  that  the  consonant 
part  of  the  middle  syllable  is  dropped,  since  the  first  two 
syllables  belong  to  the  same  note.  Similarly  g  is  the  same  as 
e  except  that  the  consonant  part  of  the  last  syllable  is 
dropped. 

20.  All  of  these  combinations  are  equal  in  time  value  and 
must  be  pronounced  in  the  same  time,  as  each  represents  the 
time  value  of  one  pulse.  Changing  the  value  of  the  pulse 
merely  changes  the  rate  of  movement  throughout  the  music. 

2  I.  Only  the  time  values  (not  the  pitch)  of  the  notes  of  the 
music  are  considered  while  tataing. 

22.  Counting  Time  consists  in  counting  the  pulses  in  each 
measure.  Whole  pulses  may  be  counted  thus,  i,  2,  etc.; 
half  pulses  thus,  i  and  2  and  etc.  ;  triplets  thus,  i  and  a  2  and 
a  etc.  ;  quarter  pulses  thus,  i  a  and  a  2  a  and  a  etc.  (the  words 
borrowing  their  time  from  the  counts).  The  shortest  note  in 
the  music  usually  determines  the  manner  of  counting,  and  the 
music  is  then  counted  through  (from  beginning  to  end)  accord- 
ingly. 

23.  Measures.  The  common  method  of  reckoning  the 
measures  in  music  is  from  measure  bar  to  measure  bar,  and 
when  the  music  begins  and  ends  with  an  incomplete  measure, 
the  two  ends  count  together  as  one  measure;  but,  evidently, 
the  two  ends  cannot  belong  to  the  same  measure. 

24.  It  is,  perhaps,  more  correct  to  reckon  the  measures- 
from  like  phase  to  like  phase  of  the  rhythm,  beginning  with  the 
first  note  of  the  music.  (If  we  measured  music  as  we  measure 
anything  else,  we  would  begin  at  one  end  and  a[)ply  the  meas- 
ure.) There  are  thus  no  incomplete  measures  at  the  begin- 
ning and  the  end  of  the  music,  for  the  music  always  begins 
and  ends  at  the   same  phase  of  the  rhythm.      The  measure 


26  MUSICC'tLOOV 

bar  is  used  primarily  for  the  purpose  of  indicating  the  princi- 
pal recurring  accent,  and  may  be  in  the  middle  or  an}-  other 
part  of  the  measure,  depending  on  the  phase  with  which  the 
music  begins.  However,  the  common  method  is  more  con- 
venient, as  it  appeals  directly  to  the  eye,  and  for  this  reason 
is  generally  adopted. 

25.  In  dividing  the  music  into  measures,  the  composer 
would  naturally  (by  singing  or  feeling  the  music)  decide 
whether  some  accents  were  stronger  than  others,  and  thus 
determine  the  principal  recurring  accent  and  place  a  bar  be- 
fore each  note  upon  which  it  fell.  He  thus  divides  the  music 
visibly  into  sections  which  ma}'  be  called  measures.  He  then 
determines  the  number  of  pulses  and  the  location  of  the  lesser 
accents,  if  any,  in  each  section ;  and  he  also  decides  the  time 
value  of  the  pulse  note  to  correspond  to  the  value  of  the  writ- 
ten note  or  notes  sung  to  each  pulse.  The  result  is  expressed 
in  the  form  of  a  fraction  and  put  at  the  beginning  of  the  music — 
it  being  understood  that  every  second  or  third  pulse  (accord- 
ing as  the  rhythm  is  even  or  uneven)  is  accented  strong  or  weak. 

26.  Determining  which  are  principal  and  which  secondary 
accents  is  largely  a  matter  of  judgment.  In  march  and  in 
dance  music  the  rhythm  is  so  pronounced  that  there  is  no 
difficulty  in  determining  the  character  of  the  accents ;  but  in 
some  other  music  the  distinction  between  the  principal  and 
the  secondary  accents  is  so  slight  that  there  is  room  for  a  dif- 
ference of  judgment,  and  different  persons  would  divide 
the  music  differently.  Thus,  whether  a  piece  of  music  is 
written  in  |  or  |  time,  or  in  |  or  |  time  (as  the  case  may  be), 
depends  on  whether  every  or  every  other  accent  is  regarded 
as  the  principal  recurring  accent.  But,  as  a  rule,  the  whole 
matter  is  left  to  the  judgment  of  the  composer. 

27.  Singing  very  slow  tends  to  develop  accents  not  other- 
wise observed,  and  at  the  same  time  the  ear  tends  to  divide 
each  pulse  into  two,  thus  changing  ",  into  ■*-  time,  etc.  Sing- 
ing very  fast  has  the  reverse  effect. 


ELEMENTARY  2/ 

28.  Tempo  or  Rate  of  Movement.  The  time  values  as  in- 
dicated by  the  shapes  of  the  notes  are  only  relative  (not  ab- 
solute). The  same  tune  may  be  sung  as  slow  or  as  fast  as  the 
singer  desires ;  but  the  rate  of  movement  with  which  the 
music  begins  should  be  maintained  throughout,  except  where 
the  movement  is  purposely  accelerated  or  retarded  for  a 
special  effect.  However,  the  time  values  given  to  the  written 
notes  are  intended  to  indicate  the  rate  of  movement  within 
reasonable  limits ;  for,  evidently,  if  the  composer  intended 
the  music  to  be  sung  fast  he  would  give  shorter  values  to  the 
notes  than  if  he  intended  it  to  be  sung  slow.  The  actual 
time  value  of  the  pulse  necessarily  depends  on  the  rate  of 
movement,  so  that  the  rate  of  movement  is  also  indicated 
within  reasonable  limits  by  the  value  given  to  the  pulse  note 
in  the  fraction  at  the  beginning  of  the  music. 

29.  It  is  desirable  to  have  a  standard  of  time  value  with 
which  to  compare.  The  standard  time  value  given  to  the 
quarter-note  is  one  second  ;  therefore  the  quarter  as  a  pulse 
note  usually  represents  about  one  pulse  per  second,  or  60  per 
minute,  written  M,  60.  [The  Metronome  (M.)  is  an  instru- 
ment for  counting  pulses.]  If  the  music  is  intended  to  be 
sung  nearer  M.  120  than  M.  60,  the  eighth-note  would  be 
taken  as  the  pulse  note  and  the  music  written  accordingly. 
If  the  music  is  intended  to  be  sung  nearer  M.  30  than  M.  60, 
the  half-note  would  be  taken  as  the  pulse  note  and  the  music 
written  accordingly. 

30.  Applying  Words  to  Music.  In  applying  the  words 
of  a  song  to  the  music,  one  syllable  is  applied  to  each  note 
unless  two  or  more  notes  are  tied  together.  Notes  may  be 
tied  by  having  their  hooks  united,  or  by  the  tie  (- — -)  or  slur 
{^ — ).  One  syllable  is  usually  applied  to  all  the  notes  thus 
tied. 

31.  Poetry  as  well  as  music  has  regularly  recurring  accents, 
producing  duple  or  triple  rhythmical  movement.  Rhythm 
therefore  is  an  essential  part  of  both  poetry  and  music.      The 


28  MUSICOLOGY 

blending  of  poetry  and  music  in  songs  is  due  largely  to  their 
conmion  rln-thniical  nature. 

32.  The  pulses  of  the  music  are  all  of  the  same  length — 
being  regulated  by  the  value  of  the  pulse  note  regardless  of 
the  written  notes  being  of  different  lengths,  some  greater 
than  one  pulse  and  others  less  than  one  pulse. 

T,],.  The  rhythm  of  the  poetry  is  not  thus  restricted.  But 
as  one  note  is  applied  to  each  syllable,  therefore  the  time 
value  of  each  syllable  depends  on  the  value  of  the  note  ap- 
plied to  it ;  so  that  a  syllable  may  cover  more  than  one  pulse 
of  the  music,  or  a  pulse  may  cover  more  than  one  syllable. 
Therefore  the  duple  or  triple  character  of  the  rhythm  of  the 
music  is  not  necessarily  determined  by  the  duple  or  triple 
character  of  the  rhythm  of  the  poetry;  each  rhythm  restrict- 
ing the  other  only  to  the  extent  of  causing  accented  syllables 
to  come  on  accented  pulses  and  unaccented  syllables  on  un- 
accented pulses. 

34.  Interrupted  Rhythms.  The  regular  rhythm  of  the 
music  is  often  purposely  interrupted  for  a  short  time  in  order 
to  heighten  by  contrast  the  rhythmical  sense  of  the  music. 
There  are  three  principal  ways  of  doing  this:  first,  by  tem- 
porarily quickening  or  retarding  the  time — indicated  by  such 
words  as  accelerando,  ritardando,  etc.  ;  second,  by  stopping  the 
rhythm  for  a  short  time — indicated  by  a  pause  ( '^  )  over  a 
note,  which  shows  that  it  may  be  prolonged  at  pleasure,  us- 
ually about  double  the  regular  time  of  the  note  (in  beating 
time  the  hand  should  be  held  during  the  pause,  as  the  rhythm 
stops);  third,  by  temporarily  displacing  the  regular  accent — 
called  Syncopation. 

35.  The  principal  ways  of  syncopating  are: 

1.  Beginning  a  note  on  an   unaccented   part  of  a  meas- 
ure and  extending  it  over  the  following  accent,  as  at  a 

(J'ig-  4) ; 

2.  Reversing  the  accent  by  placing  accent   marks  over 
the  usually  unaccented  parts  of  a  measure,  as  at  b\ 


ELEMENTARY  29 

3.  Writing  rests  on  the  accented  parts  of  a  measure  and 
notes  on  the  unaccented  parts,  as  at  c; 

4.  Tying  unaccented  notes  to  accented  notes,  as  at  d\ 

5.  Writing    notes    between,    instead    of    on,    the    usual 
pulses  of  a  measure,  as  at  c. 

abbe  d         d  e 


.^ggiggEg^g^jgiJ: 


Fig.  4. 

36.  Triplets.  A  numeral  placed  above  or  below  a  group 
of  notes  affects  their  normal  value.  The  most  common  in- 
stance is  a  3  placed  above  or  below  a  group  of  three  notes, 
which  indicates  that  they  are  to  be  sung  or  played  in  the  time 
of  two.  The  notes  thus  affected  are  called  triplets.  They 
do  not,  however,  affect  the  rhythm  of  the  music. 

37.  Repeats.  Dots  between  the  lines  at  the  left  of  a  bar 
mean  to  repeat,  and  dots  at  the  right  of  a  bar  show  zvJiere  a 
repeat  begins.  D.  C.  {Da  Capo)  means  repeat  frojii  the  begin- 
niiig.  D.  S.  {Dal  Segno)  means  return  to  the  sig)i  (S  or  -S'). 
The  word  fine  means  the  end  of  a  repeat. 

EXPRESSION 

1.  Expression  marks  affect  the  character  of  music  as  regards 
Power,  Quality,  and  Movement,  for  the  purpose  of  bringing 
out  the  sentiment. 

2.  Terms  relating  to  Power.  Pianissimo  {pp),  very  soft; 
Piano  (/),  soft;  Mezzo  {m),  medium;  Forte  {/),  loud;  For- 
tissinio  {ff),  very  loud  ;  Crescendo  (— ==^  ),  increasing  tone  or 
tones  ;  Dinmuiendo  { I^:^==~  ),  diminishing  tone  or  tones  ;  Sivell 
{  ^=^^^:==-  ),  increasing  and  diminishing;  Forzando  {fz,  or>-), 
a  sudden  burst  of  tone;  Staeeato  (im  ),  short  and  distinct; 
Semi-Staeeato  {  •••  ),  less  short  and  distinct;  Legato  (opposite 
of  staccato),  smooth  and  connected — sometimes  indicated  by 
a  slur  (x*— )  placed  over  the  notes. 


30  MUSICOLOGY 

3.  Terms  relating  to  Quality.  Somber,  tones  of  reverence, 
sadness,  or  fear;  Clear,  tones  of  courage,  joyfulness,  or  gay- 
ety ;  Maestoso,  loud  and  majestic ;  Affettiioso,  soft  and  sad ; 
Dolce,  soft  and  sweet ;  Con  Spirito,  with  spirit ;  Con  Dolore, 
with  grief;    Giojoso,  joyfully. 

4.  Terms  relating  to  Movement  (or  tei/ipo).  The  chief  of 
these  from  slowest  to  fastest  are :  Grave,  Largo,  LargJietto, 
Adagio,  Lento,  Andante,  Andantiuo,  Moderato,  Allegretto, 
Allegro,  Presto,  Prestissimo. 

5.  For  other  musical  terms  see  Musical  Dictionary  at  back 
of  book. 

6.  Embellishments.  A  Turn  (««)  placed  over  a  note 
iz?:z)  means  that  it  is  to  be  sung  or  played  thus.    *  *  •  ^   > 

involving  the  notes  above  and  below,  beginning  with  the  high- 
est.    An   Lnverted  Turn  {^o  ox%)   includes  same  notes  in  an 

inv^erted  order.      A  turii  after  a  note  \  — »—  )  or  dotted   note 

( ifzii )  is   made  by   striking   the   note   first   antl   making   the 

turn  afterward.  A  sharp  or  flat  above  the  turn  affects  the 
upper  auxiliary  note;  but  if  below,  it  affects  the  lower  aux- 
iliary note. 

7.  A  Trill  (tr)  placed  over  a  note  means  a  rapid  alterna- 
tion with  the  note  above  to  the  time  value  of  the  note.  A 
trill  usually  ends  with  a  turn,  and  may  also  begin  with  a  turn. 
There  are  also  trill  chains  (succession  of  trills),  and  double 
trills  (two  trills  at  the  same  time). 

8.  The  Mordant  (w)  is  a  short  trill,  involving  usually  but 
one  beat. 

9.  These  signs  are  used  to  abbreviate  the  writing  of 
music : 

10.  A  line  drawn  through  any  sign,  thus,  f^,  inverts  it. 

11.  Any  two  signs  combined  give  the  combined  effect  of 
both ;   thus  iw,  or  ^,  is  a  mordant  followed  by  a  ///;-;/. 


ELEMENTARY  3 1 

12.  A  Grace  Note  is  a  small  note  written  before  or  after 
the  note  it  is  intended  to  embellish  and  from  which  it  bor- 
rows its  time. 

13.  The  Appoggiatura  (usually  written  as  a  grace  note)  is 
an  embellishing  note  prefixed  to  an  essential  note  on  an  ac- 
cented pulse  of  a  measure.  When  written  as  a  grace  note,  it 
receives  the  accent  and  borrows  half  or  more  of  the  time  value 
of  the  note,  and  is  therefore  made  more  prominent  than  the 
note  to  which  it  is  prefixed.  Its  effect  is  languishing  and 
sorrowful. 

14.  The  Acciaccatnra  is  a  grace  note  a  half-step  above  or 
below  the  note  to  which  it  is  prefixed.  It  is  very  short  and 
never  accented.  It  takes  as  little  of  the  time  value  of  the 
note  to  which  it  is  prefixed  as  possible  (opposite  of  the  appog- 
giatura).    Its  effect  is  crisp,  bright,  and  joyous. 

15.  An  After  Tone  is  a  grace  note  which  follows  an  essen- 
tial note. 

KEYS 

1.  Music  is  divided  into  what  are  called  Keys — tones  hav- 
ing a  sympathetic  relation  to  each  other.  A  key  therefore 
is  a  family  of  related  tones.  Every  key  is  made  up  of  seven 
steps,  or  intervals,  which  are  repeated  in  each  octave. 

2.  See  Chart  I.  at  front  of  book.  Place  key  patterns  in 
position  (see  instructions  on  Chart). 

3.  The  lines  on  the  face  of  the  Chart  represent  the  lines 
of  the  music  staff,  and  the  dotted  lines  represent  the  spaces 
of  the  music  staff.  These  are  marked  with  the  scale  letters 
A,  B,  C,  D,  E,  F,  G,  and  are  so  spaced  as  to  represent  the 
natural  tone  steps,  or  major  key  intervals;  these  seven  letters 
are  repeated  in  each  octave.  Observe  that  there  is  a  half- 
step  between  each  E  and  F,  and  B  and  C,  and  whole  steps 
between  all  the  other  letters.  The  short  marks  show  the 
half-steps  between  the  letters.  Each  short  mark  is  the  sharp 
(«)  of  the  letter  below  it  or  the  fiat  (I?)  of  the  letter  above  it. 

4.  Next    observe     that     the    movable    key    strip   (marked 


32  MUSICOLOGY 

major)  also  has  lines  drawn  across  it,  which  are  also  spaced  to 
represent  the  major  key  intervals,  or  tone  steps,  and  are 
marked  with  the  seven  syllables  Do,  Re,  Mi,  Fa,  Sol,  La, 
Ti,  and  the  seven  numerals  i,  2,  3,  4,  5,  6,  7,  which  are  re- 
peated in  each  octave.  Observe  also  that  there  is  a  half- 
step  between  each  Mi  and  Fa  (or  3  and  4),  and  each  Ti  and 
Do  (or  7  and  i). 

5.  There  is  a  natural  sympathy  existing  between  the  tones 
thus  related.  Any  group  of  tones  thus  related  is  called  a 
Jiiajor  key.  This  major  key  strip  may  therefore  be  regarded 
as  the  pattern,  or  mode  (meaning  mode  of  arrangement),  for 
all  the  major  keys. 

6.  There  is  also  a  minor  key  pattern,  or  mode  (see  key 
strip    marked  minor).      Observe   that   there    is  a  half-step  be- 

.|  tween  each  2  and  3,  5  and  6,  and  7  and  i,  and  an  augmented 
\  interval  (equal  to  one  and  one-half  steps)  between  6  and  7. 
Sol  being  sharped  is  changed  to  si,  and  the  key-note  (i)  is  on 
La  instead  of  Do.  It  is  formed  by  sharping  5  (Sol)  of  the 
major  key  pattern  and  changing  the  key-note  to  La.  This 
is  the  pattern,  or  mode,  for  all  minor  keys. 

7.  The  major  and  minor  key  patterns  are  placed  side  by 
side  in  Fig.  10  (p.  43)  for  comparison.  The  major  mode  is  of 
major  (greater)  importance,  and  the  miiior  mode  is  of  minor 
(less)  importance. 

8.  To  adjust  Chart  I.  to  the  key  in  which  any  piece  of 
music  is  written,  first  count  the  sharps  (  #'s)  or  flats  (  !? 's)  at 
the  beginning  of  the  music  (the  signature),  then  find  the  same 
number  in  the  key  table  at  top  of  Chart :  the  letter  opposite 
will  be  the  key  letter. 

9.  Now  set  the  major  key  pattern  so  that  tiic  kej'-note  i 
(Do)  will  be  opposite  the  key  letter.  The  syllables  on  the 
key  pattern  will  match  all  the  letters  on  the  staff  except  the 
letters  which  are  to  be  sharped  (raised  a  half-tone)  or  flatted 
(lowered  a  half-tone).  Observe  also  that  the  sliarps  or  flats 
in  the  signature  of  any  piece  of  music  are    on  the  same  lines 


ELEMENTARY 


33 


and   spaces  that   the   key  pattern  (wlien  set   to  the  same  key) 
shows  are  to  be  sharped  or  flatted. 

10.  Take  a  music  book  and  test  the  signatures  in  all  the 
different  keys  by  Chart  I.,  by  setting  the  major  key  pattern 
to  the  desired  key  and  comparing  the  signature  with  it. 

1 1.  When  a  piece  of  music  has  no  signature  it  is  in  the  key 
of  C.  Now  set  the  key  pattern  so  that  i  (Do)  will  be  oppo- 
site C.  It  will  be  seen  that  all  the  letters  and  syllables  match, 
so  that  none  of  the  letters  have  to  be  sharped  or  flatted  to 
conform  to  the  key  pattern.  For  this  reason  the  key  of  C  is 
also  called  the  natural  key. 

12.  It  will  also  be  noticed  from  the  Chart,  that  when  any 
letter  is  sharped  or  flatted  it  is  affected  the  same  way  in  each 
octave.  Therefore  the  sharps  or  flats  in  any  signature  affect 
the  same  letters  in  each  octave.  It  is  customary  to  place  the 
signature  once  on  each  staff. 

13.  Set  the  major  key  pattern  to  the  keys  of  one,  two,  three, 
four,  five,  six,  and  seven  sharps  in  succession  (writing  down  the 
letters  sharped  in  each  case).  Observe  that  each  key  is  formed 
by  adding  one  new  sharp  to  those  of  the  preceding  key. 

14.  Examine  the  flat  keys  in  the  same  way. 

15.  As  each  new  key  is  formed,  the  last  sharp  or  flat  is 
placed  in  the  signature  on  the  line  or  space  affected,  a  little 
in  advance  to  the  right  of  the  others. 

16.  Fig.  5  shows  the  signa- 
tures of  the  key  of  seven  sharps 
and  the  key  of  seven  flats.  These 
include  all  the  other  signatures, 
by  taking  them  from  left  to 
right  as  they  come. 
17.  The  intermediate  tones,  between  the  fixed  tones  rep- 
resented by  the  letters  of  the  staff,  are  indicated  on  the  Chart 
by  the  short  marks  between  the  letters.  They  are  indicated 
in  music  by  sharps  and  flats.  They  arc  indicated  on  the  key- 
board of  the  piano  or  organ  by  the  black  keys. 


Fig.  5. 


34 


MUSICOLOGY 


Fig.  G.    -jjy 


•r_" 


1 8.  Fig.  6  shows  the  relation   between  the  music  staff   and 

the  piano  or  organ  key-board. 
Observe  that  the  letters  on  the 
staff  match  the  white  keys  of 
the  key-board,  and  the  short 
marks  between  the  letters  match 
the  black  keys.  Each  black 
key  is  the  sJiarp  of  the  white 
key   next    below,  or  the  fiat   of 

y — s- ft — -1^    I       the     Avhite     key      next      above. 

r^  j» "  p """"-— ^        (When     referring     to    the   key- 

\^_y  "E— "iLai — I       \^Q-^x<\,   the  word    key    refers  to 

one  of  the  white  or  black  finger- 
bars  ;  elsewhere  the  word  key 
refers  to  a  family  of  related 
tones.) 

19.  See  Chart  II.  at  back  of  book.  Place  key  pattern  in 
position  (see  instructions  on  Chart). 

20.  A  general  knowledge  of  music  involves  also  a  knowl- 
edge of  the  principles  of  the  key-board.  It  is  only  necessary 
to  remark  here  that  all  observations  on  Chart  I.  apply  also  to 
Chart  II.  in  a  similar  manner. 

21.  As  the  signature  of  any  piece  of  music  shows  what 
tones  are  to  be  sharped  or  flatted  throughout  the  music,  and 
as  the  black  keys  represent  sharps  or  flats,  therefore  the  sig- 
nature of  any  piece  of  music  shows  what  black  keys  are  used 
in  playing  it.  As  there  are  only  five  black  keys  in  each  oc- 
tave, therefore  six  or  seven  sharps  or  flats  in  the  signature 
show  that  E  and  H  or  V  and  C  are,  one  or  both,  sharped  or 
flatted,  as  the  case  may  be.  But  as  there  is  no  black  key 
between  E  and  F  or  B  and  C  (they  being  only  a  half-step 
apart),  one  is  used  as  the  flat  or  sharp  of  the  other. 

22.  A  Diatonic  Scale  consists  of  the  regular  tones  of  a  key 
in  successive  order,  as  shown  by  the  major  and  minor  key 
patterns. 


ELEMENTARY  35 

23.  A  Chromatic  Scale  consists  of  both  the  regular  and  in- 
termediate tones  in  successive  order  (see  Fig.  2,  p.  21). 

24.  Although  in  music  the  lines  of  the  staff  (and  therefore 
the  letters)  are  spaced  equally,  yet  it  must  be  fixed  in  the 
mind  that  there  is  always  a  half-step  in  tone  between  E  and 
F,  and  between  B  and  C,  and  a  whole  step  between  all  the 

^  other  letters  as  shown  on  the  Charts  ;  so  that  the  key  of  C  con- 
forms to  the  pattern  without  any  sharps  or  flats.  In  all 
other  keys,  the  sharps  or  flats  in  the  signature  show  what 
lines  and  spaces  must  be  raised  or  lowered  in  tone  to  make 
the  key  conform  to  the  pattern.        * 

READING   MUSIC 

1.  The  first  step  is  learning  the  letter  names  of  all  the  lines 
and  spaces  of  both  the  bass  and  treble  staffs  so  that  they 
can  be  named  without  hesitation.  The  second  step  is  learn- 
ing to  run  the  scale  with  the  voice,  using  the  key  syllables. 
Do,  Re,  Mi,  Fa,  Sol,  La,  Ti,  Do :  this  will  have  to  be  learned 
from  some  musical  instrument  or  the  voice  of  a  teacher. 

2.  Each  of  the  key-tones,  represented  by  the  seven  syl- 
lables, produces  a  certain  mental  effect.  The  character  of  the 
key-tones  may  be  described  as  follows:  Do,  "the  strong  or 
firm  tone";  Re,  "the  rousing  or  hopeful  tone";  Mi,  "the 
steady  or  calm  tone";  Fa,  "the  desolate  or  awe-inspiring 
tone";  Sol,  "the  grand  or  bright  tone";  La,  "the  sad  or 
weeping  tone  " ;  Ti,  "  the  piercing  or  sensitive  tone."  The 
mental  effect  of  any  tone  is  called  its  key-tonality  and  is  due 
to  its  relation  to  the  key-note. 

3.  After  learning  the  key-tonality  of  each  tone  the  key-note 
may  be  pitched  with  the  voice  to  any  key,  and  the  other  tones 
will  adjust  themselves  to  it  by  reason  of  their  key-tonality. 
The  key-tonality  of  each  tone  does  not  change ;  it  is  only  the 
pitch  that  changes.  This  may  be  illustrated  by  shifting  the 
key  pattern  (Chart  I.)  to  bring  Do  opposite  any  letter:  it  is 
evident  that  the  syllables  all  move  together  and  that  their 
relation  to  each  other  and  to  Do  does  not  change. 


36  MUSICOLOGY 

4.  A  piece  of  music  (being  made  up  of  key-tones)  can  be 
changed  to  any  key  just  the  same  as  the  key  pattern  can  be 
shifted  to  any  key  (being  merely  a  change  of  pitch),  but  the 
music  should  be  written  in  that  key  which  will  bring  all  the 
notes  within  the  average  range  of  the  voice  as  shown  by  the  staff. 

5.  Changing  the  key  of  a  piece  of  music  is  called  Trans- 
position. We  may  transpose  any  piece  of  music  to  suit  the 
voice,  as  the  key  is  only  a  question  of  pitch. 

6.  There  can  be  no  intelligent  understanding  of  music  until 
this  key-tonality,  or  sympathetic  relation  of  the  tones  of  any 
key,  is  understood  ;  and  this  is  most  readily  accomplished  by 
the  use  of  the  key  syllables. 

7.  The  mental  effect  (key-tonality)  of  a  tone  is  due  to  a 
sort  of  unconscious  mental  measurement  of  the  interval,  or 
distance,  from  the  point  of  repose  (key-note  previously  es- 
tablished in  the  mind),  and  its  association  with  the  tone  or 
tones  immediately  preceding,  the  effect  of  which  has  not  yet 
passed  out  of  the  mind.  A  tone  entirely  unassociated  with 
other  tones  has  no  key-tonality  by  which  its  key  can  be  rec- 
ognized, as  it  may  belong  to  different  keys. 

8.  The  ability  to  recognize  the  key-tonality  of  tones  (and 
also  rhythm)  is  so  developed  through  practice  as  to  enable 
one  to  mentally  hear  a  piece  of  music  by  merely  following 
the  notes  with  the  eye,  as  distinctly  as  if  it  were  sung  or 
played  ;  and  also  to  mentally  see  a  piece  of  music  (how  it  is 
written)  by  merely  hearing  it  sung  or  played,  and  thus  be 
able  to  write  it. 

9.  The  ability  to  thus  hear  with  the  eye  is  essential  in 
order  to  read  music  readily,  while  the  ability  to  see  with  the 
ear  is  essential  in  order  to  write  music;  both  are  the  natural 
growth  of  practice,  exactly  as  in  reading  and  writing  spoken 
language.  We  understand  what  we  read  without  reading 
aloud,  thus  applying  the  principle  of  hearing  with  the  eye. 
In  writing  what  wc  hear  spoken,  we  apply  the  principle  of 
seeine  with  the  car. 


ELEMENTARY  n 

10.  Beating  Time  is  indicating  the  pulses  of  the  music 
with  the  hand. 

1 1.  Tataing  the  music  is  singing  the  time  names  (p.  24 :  17, 
18)  of  the  notes  on  one  tone.  This  is  more  especially  for  the 
purpose  of  fixing  in  the  mind  the  relative  time  value  of  the 
parts  of  broken  pulses,  after  which  beating  time  is  sufficient. 
Both  exercises  are  intended  to  develop  the  faculty  of  feeling 
the  rhythm  of  music. 

12.  Solfaing  the  music  is  singing  the  music  with  the  key 
syllables. 

13.  Laing  the  music  is  singing  the  music  with  the  open 
syllable  La,  the  object  of  which  is  to  overcome  the  depend- 
ence on  the  key  syllables  which  solfaing  tends  to  produce. 

14.  These  are  the  usual  stages  of  practice  in  learning  to 
read  music,  before  applying  the  words.  Beating  time  and 
solfaing  are  the  essential  stages,  the  others  are  side  helps. 

THE   TONIC   SOLFA   NOTATION 

1.  There  are  two  principal  methods  of  notating,  or  writing 
music,  in  use. 

2.  The  Staff  Notation  consists  in  representing  the  music  by 
notes  on  a  staff,  as  already  explained. 

3.  The  Tonic  Sol/a  Notation  consists  in  representing  the 
music  by  the  first  letters  of  the  key  syllables  (instead  of  notes) 
written  in  a  straight  line.  The  first  octave  above  the  key- 
note (called  the  unmarked  octave)  is  written  thus — d  r  PI  f  S 
1  t;  the  octave  above  the  unmarked  octave,  thus — d'  r'  Hi'  f 
S'  1'  t'  ;  the  octave  below  the  unmarked  octave,  thus — d|  Ti 
Wi  f|  S|  1|  t|.  The  figure  2  may  be  placed  above  or  below  if 
a  further  extension  is  needed  in  either  direction. 

4.  The  time  is  indicated  by  punctuation  marks  between  the 
letters.  A  bar  (  I  )  is  placed  before  each  principal  accented 
pulse,  thus  dividing  the  music  into  measures.  A  short  bar 
(  I  )  is  placed  before  each  lesser  accented  pulse,  and  a  colon 
(:)  before  each  unaccented  pulse,  thus  dividing  each  meas- 
ure into  pulses.      A  period  ( .  )  divides  a  pulse  into  halves;  a 


38  MUSICOLOGV 

comma  (,  )  divides  a  half-pulse  into  quarters;  a  period  and 
comma  close  together  divide  a  pulse  into  three-quarters  and 
one-quarter — the  period  being  on  the  side  of  the  greater  part.* 
Inverted  commas  divide  a  pulse  into  thirds.  A  macron  ( — ) 
indicates  a  continuance  of  the  preceding  tone.  Rests  are  in- 
dicated by  vacant  spaces.  A  horiJ:ontal  bar  ( — )  over  or 
under  two  or  more  letters  ties  them  together  the  same  as  a 
tie  ('    ")  or  slur  ('^)  in  other  music. 

5.  The  time  valufe  of  each  pulse,  or  the  rate  of  movement, 
is  usually  shown  by  a  metronome  mark,  thus,  M.  60  (or  any 
other  number),  placed  over  the  beginning  of  the  music.  The 
key  is  also  shown  by  the  key  letter  being  placed  over  the  be- 
ginning of  the  music.  Also,  whenever  the  key  changes  dur- 
ing the  music  (modulation,  p.  90)  the  new  key  letter  is  placed 
over  the  point  where  the  new  key  begins,  and  ihe^  Bridge  To?ie 
has  its  old  name  placed  above  its  new  name,  thus,  ^.  and  takes 
its  new  name  at  its  old  pitch,  and  the  following  tones  in  the 
new  key  are  pitched  accordingly. 

6.  The  following  example  is  intended  merely  to  illustrate 
each  of  the  foregoing  principles: 


Key  of  D.         M.  60 

A.t. 

1  d        :i<i  .f   Is 

:d'.t,l  1  s,l.t,d':t.,l    1  s 

:^d 

1  l|     :d,.r  InkFcd:!, 

Id     Ids    If     :      In    :- 

Fu,.  7. 

.r|d  : 

7.  The  small  letter  beside  each  new  key  letter  above  is  the 
{^  or  ft )  tone  which  distinguishes  the  new  key,  and  is  placed 
before  or  after  the  key  letter,  according  as  it  is  i'  or  j|. 

INTERRELATIONSHIP    OF    MAJOR    KEYS 

I.  To  study  the  major  keys,  set  the  major  key  pattern 
(Chart  I.')  first  to  the  key  of  C  and  notice  that  V  is  opposite 
the  figure  4  on  the  pattern  ;  now  set  the  pattern  to  the  key 
of  one  sharp  and  notice  that  F  (which  was  the  4th  of  the  old 


ELEMENTARY  39 

key)  is  sharped  and  becomes  the  7th  of  the  new  key.  Now 
notice  the  4th  of  this  key  and  move  to  the  key  of  two  sharps, 
and  we  again  find  that  the  4th  of  the  old  key  (sharped) 
becomes  the  7th  of  the  new  key.  Continue  this  operation 
through  all  the  sharp  keys.  Thus  we  find  that  all  the  sharp 
keys  are  formed  by  sharping  the  4th  letter  of  the  preceding 
key,  which  then  becomes  the  7th  of  the  new  key.  Observe 
also  that  the  key-note  (i)  of  each  key  is  on  5  of  the  ]:)reced- 
ing  key. 

2.  Study  X\\Q  Jlat  keys  by  setting  the  pattern  again  to  the 
key  of  C,  and  notice  that  B  is  opposite  7  ;  now  set  the  pat- 
tern to  the  key  of  one  flat  and  notice  that  B  (which  was  the 
7th  of  the  old  key)  is  flatted  and  becomes  the  4th  of  the  new 
key.  Now  flat  the  7th  of  this  key,  and  move  to  the  key  of 
two  flats  (Bt' — set  i  on  the  short  mark  below  B) ;  the  letter 
that  was  flatted  will  now  be  the  4th  of  the  new  key.  Con- 
tinue through  all  the  flat  keys.  Thus  we  find  that  \h.t  fiat 
keys  are  formed  by  flatting  the  7th  of  the  old  key,  which  be- 
comes the  4th  of  the  new  key  (reverse  of  sharp  keys).  Ob- 
serve also  that  the  key-note  (i)  of  each  key  is  on  4  of  the 
preceding  key. 

3.  The  last  sharp  is  always  on  the  7th  of  the  key,  and 
controls  the  half-step  between  7  and  I.  The  last  flat  is 
always  on  the  4th  of  the  key,  and  controls  the  half-step 
between  3  and  4. 

4.  Observe  that  the  half-steps  between  each  3  and  4  and 
each  7  and  i  of  the  major  key  pattern  divide  it  into  alternate 
2\  and  31^  step  sections;  and  observe  also  that  raising  each  4 
or  lowering  each  7  a  half-step  would,  in  either  case,  reverse 
the  sections.  This  is  exactly  what  happens  to  any  key  when 
we  sharp  the  4th  or  flat  the  7th  of  the  key.  As  the  key  pat- 
tern cannot  be  changed,  it  will  have  to  be  moved  up  or  down 
till  the  sections  correspond,  thus  forming  a  new  key,  since 
the  key-note  (i)  which  determines  the  key  has  been  moved  to 
a  new  position. 


40  MUSICOLOGY 

5.   These  observations  on   the  interrelationship   of    major 
keys  may  be  put  into  the  form  of  a  table,  as  shown  in  Fig.  8. 

7G5        4321  1234         5         67 

Key  of  b:^  C''     G''     D!?     a*'      E''      B''      F       C      G       D       a       E       B       F«    C«' 

-F b4;:r--=^7 3 6 2 5 riTrmir^— '#7 J3 $Q J3 J5 ^zTzrli- 


E       b3       be       b2       b5       bi ^b4,--'''7        3        6        2        5       J i-'-'irr       P 

D b2 b5 btzirH^'"'"' 3 6 2 5 tr-Tr-Jb--^Sv' Jt3 JG $2- 

C  bl___b4,^-"'7  73  6  2  5  1 i-''''fT         $3         #6         {2         J5         |1 

B bT -"bs be b2 bs btmrbi  ---'l^ ^3 e §: 5 j iT^^-^ii:- 

A       be       b2       bs       bi  __  b4,--'"r        3        6        2        5        i i-'-'Ifr       js       jfe 

G b  5 — -bi — -b4,--'"r ^3 6 2 5 1 t^-^'IF S3 jte 1^ $5- 

F         b4.^'''7  3  6  2  5  i     __  4,-"lf7         53         $6         J2        $5         $1         $4 

b3 be b2 bo bi-^:^47^-=^ 3 e 2 5 1^ 4^.-^l67: — ^S3- 


D        b3         bs         bl_     .N,---'"?  3  6  2  5  1_       _i,-'''«7         $3         $6         S2 

— C bi-  _-b4---- — 7-~ '— 3-  -6-  -a—  -5—  .-1—  jii-,— «-'ffr-"-$3 — jfe-  -$2-   -jjs — jfi- 

B         b7         &3         be         b2         bs         bl_       b4,-'''7     "36  25  J i^'''S~ 

-A be b2 bs bi b4,--  -'"7        3 6 2 5 -i—^n^h^-^V P J6- 

G         bs         bl _bjl.'--''7       '36  25  1__       i^'-'V        f3         $6         52        JS 

-F b4;;r-^7 — ^3 6 2 5 i-rzzrij^^-^f^-^lt3 S6 J2 $5 H— T-ii- 

E       b3       be       b2       bo       bi b4  ^-'T        3        6        2        5        1. 4,'''fr      l3 

-D — b2 bs bl— rz-bi,---'! ^3" e 2 5 1_  _   j^—^iP" fi3 jte js- 

/C         bl._  _b4,-"'7'         3  6  2  5  1_       i,'-'S7"    'P         J6         «2         jp  $1 

-B — b7 — ^3 be— b2 bs bii — ibtz^^-^T ^3 e 2 5r= — hzz-zi^^^-^'tp- 

A        bo         b2         E»5  '       bl bi-'-'  7  3  6  2  5  1__     4^'''S7         J3         JO 

-G bs- — bl— b4^^--^-rf- 3- io- — zH- — § i- i^^th—^m- — je p «5- 

FiG.  8.     Major  Key  Table. 

6.  In  this  table  the  figures  in  each  key  correspond  to  the 
figures  on  the  key  pattern  and  also  represent  the  syllables 
(the  syllables  being  omitted  for  the  sake  of  simplicity).  Ob- 
serve that  here  the  staff  is  spaced  equally  as  in  music,  instead 
of  as  on  Charts  I.  and  II.  ;  the  sharps  and  flats  instead  in- 
dicating the  proper  intervals  in  each  key  (half-steps  being 
understood  between  each  E  and  F,  and  B  and  C).  The 
natural  key  of  C  is  placed  in  the  middle  with  the  sharp  keys 
on  the  right  and  the  flat  keys  on  the  left. 

7.  Begin  at  the  extreme  left  (key  of  C[?),  and,  proceeding 
from  left  to  right,  notice  that  each  key  is  formed  by  remov- 
ing the  flat  on  the  4th  of  the  preceding  key  in  succession  till 
all  the  tones  of  the  staff  are  made  natural,  then  sharping  the 
4th  of  the  preceding  key  in  succession  till  all  the  tones  of  the 
staff  are  sharped  ;  the  4th  in  each  case  becoming  the  7th  of 
the  next  key  toward  the  right. 

S.    Now  begin  at  the   rit^ht  and    reverse,  noticing  that  each 


ELEMENTARY  4 1 

key  is  formed  by  removing  the  sharp  on  the  7th  of  the  pre- 
ceding key  in  succession  till  all  the  tones  of  the  staff  are  made 
natural,  then  flatting  the  7th  of  the  preceding  key  in  succes- 
sion till  all  the  tones  of  the  staff  are  flatted  ;  the  7th  in  each 
case  becoming  the  4th  of  the  next  key  to  the  left. 

9.  It  will  be  seen  that  a  natural  ( tj  understood  in  the  table 
but  expressed  in  music)  is  the  same  as  a  sharp  in  flat  keys, 
and  the  same  as  a  flat  in  sharp  keys. 

10.  Notice  also,  from  the  table,  that  the  key-note  (i)  of 
any  key  is  opposite  5  of  the  next  key  to  the  left,  and  oppo- 
site 4  of  the  next  key  to  the  right.  By  drawing  dotted  lines 
through  the  half-steps  (see  table)  we  readily  see  how  raising 
the  4th  a  half-step  (going  toward  the  right)  or  lowering  the 
7th  a  half-step  (going  toward  the  left),  in  either  case,  reverses 
the  sections,  causing  a  readjustment  of  the  key-note  (i), 
which  must  always  have  the  same  position  with  reference  to 
the  sections. 

11.  To  tell  the  key  of  any  piece  of  music  from  the  signa- 
ture, remember  that  the  last  sharp  (the  one  farthest  to  the 
right)  is  always  on  the  7th  of  the  key,  therefore  the  key  let- 
ter is  the  first  letter  above ;  if  this  is  too  high,  drop  down  an 
octave  to  the  same  letter  below.  Also,  the  last  flat  is  always 
on  the  4th  of  the  key,  therefore  the  key  letter  is  the  fourth 
letter  below.  It  also  happens  that  the  next  to  the  last  flat  is 
always  on  the  key  letter  itself  (see  table). 

12.  Observe  that  the  letters  at  the  top  of  the  table  are  the 
names  of  the  keys,  and  the  figures  above  the  letters  show  the 
number  of  sharps  or  flats  in  each  octave  of  that  key  (being 
the  number  of  sharps  or  flats  in  the  signature). 

13.  As  each  short  mark  between  the  letters  on  the  face  of 
Chart  I.  represents  either  the  sharp  of  the  letter  below  or  the 
flat  of  the  letter  above,  therefore  any  sharp  can  be  changed 
into  a  flat  or  a  flat  into  a  sharp  by  changing  its  letter  name, 
without  affecting  its  tone.  Thus  C|^  is  the  same  tone  as  D  t' , 
D  ^  same  as  E  i' ,  E  H  same  as  F  (or  E  same  as  F  I' ),  F  ^  same 


42 


MUSICOLOGY 


as  G  b  ,  G  H  same  as  A  1^,  A  ^  same  as  B  b  ,  and  B  |^  same  as  C 
(or  B  same  as  C^^).      This  is  called  the  EnJiarvionic  Cliange. 

14.  Set  the  major  key  pattern  to  the  key  of  C  j^  and  notice 
that  it  is  the  same  as  the  key  of  Di^,  as  the  key-note  (i)  will 
have  to  be  set  to  the  same  mark  in  either  case ;  and  it  only 
requires  the  enharmonic  change  of  each 
tone  to  pass  from  one  key  to  the  other. 
Therefore  the  keys  of  Cjf  and  Dt'  are 
equivalent  or  interchangeable  keys.  For 
the  same  reason,  the  keys  of  F^  and  G  '^ 
and  the  keys  of  B  and  C  1^  are  equiva- 
lent or  interchangeable  keys.  We  see, 
therefore,  that  although  theoretically  , 
there  are  fifteen  major  keys  yet  act-  ' 
ually  there  are  only  twelve. 

15.  Now  suppose  that  the  major  key 
table  is  cut  out  and  bent  around  in  a 
circle  (Fig.  9),  the  first  three  and  last 
three  keys  overlapping  (being  inter- 
changeable keys).  This  gives  the  key 
circle.  The  relations  of  the  keys  al- 
ready pointed  out  may  be  followed  con- 
tinuously around  the  circle  by  making  the  enharmonic  change 
at  any  one  of  the  three  sets  of  interchangeable  ke}-s. 

16.  It  will  be  seen  from  the  top  of  the  major  key  table  and 
also  from  the  table  at  top  of  Chart  I.,  that  there  are  both  a 
sharp  and  a  fiat  key  on  each  letter  of  the  staff.  Either  key 
may  be  changed  to  the  other  by  simply  changing  the  signa- 
ture without  changing  the  notes  on  the  staff,  but  the  pitch  of 
the  entire  piece  of  music  thus  changed  will  be  either  raised  or 
lowered  (as  the  case  maybe)  one  half-step  in  tone.  Thus  the 
key  of  E  (four  sharps)  may  be  changed  to  the  key  of  El'  (three 
flats)  by  changing  the  signature,  as  it  is  only  by  the  signature 
that  we  can  tell  whether  the  music  is  written  in  the  key  of  E 
or  E  k     This  is  because  in  music  the  lines  of  the  staff  are 


fc      -ft 

%-.   ^      ta      "■ 
.1    "^- .--,81-       ■, 

fe-..  ^     r^  -'-■■ 

V^        16  OX 

Fig.  9.     Key  Circle. 


ELEMENTARY 


43 


spaced  equally  and  every  note  is  either  on  a  line  or  space,  and 
sharping  or  flatting  it  does  not  change  its  position  on  the  staff. 

17.  Comparing  this  change  with  the  enharmonic  change, 
we  see  that  in  this  change  to  a  key  on  the  same  letter,  the 
names  and  therefore  the  position  of  the  notes  on  the  staff  are 
not  changed  but  the  tones  themselves  are  changed  one  half- 
step.  In  the  enharmonic  change,  the  tones  are  not  changed, 
but  their  names  and  therefore  position  on  the  staff  are  changed 
one  degree. 

18.  The  relationship  between  any  two  keys  depends  upon 
the  number  of  tones  common  to  both  Thus  if  the  signatures 
of  two  keys  differ  by  only  one  sharp  or 
flat,  the  keys  are  in  first  relationship,  as 
all  the  tones  are  common  but  one.  If 
their  signatures  differ  by  two  sharps  or 
flats,  the  keys  are  in  second  relationship, 
etc.  The  major  key  table  shows  the  order 
of  relationship  of  major  keys. 


INTERRELATIONSHIP   OF    MAJOR 
AND    MINOR    KEYS 


-1 — Do-^ 

-6 — La 

1-5— Soi- 

i—Fa-] 
3— il/r 

2 — Be 

-1 — Do 

"T—TH 

C — La- 
5—Sol- 


I.  In  Fig.  10  the  major  and  minor  Key 
patterns  are  placed  side^  by  side  for  com- 
parison. Observe  that  in  the  major  key 
pattern  there  is  always  a  half-step  be- 
tween 3  and  4  and  between  7  and  i  in 
each  octave,  and  that  in  the  minor  key 
pattern  there  is  always  a  half-step  be- 
tween 2  and  3  and  between  5  and  6  and 
between  7  and  i  and  an  augmented  or 
enlarged  interval  (equal  to  \\  steps) 
between  6  and  7.  Observe  also  that 
sharping   the  5th  (Sol)  of  the  major  key 

pattern  and  changing  the  key-note  (i)  to   La  gives  the  minor 
ke)'  pattern.      This    is  called    the   relative   minor,  because   of 


4— Fa 
-z—Mi- 


2— Re 


-Si 


-6 — Fa 
5 — Mi 


4 — Re 


-1 — Do 

K — Ti~ 

6 — La 
5—Sol-\ 

i—Fa 
3 — Mi- 

•2 — Re- 

-1 — Do 


3 — Do 
2 — Ti-\ 


-1 — La 

-; Sir 


6— JT» 
5 — Mti 


4 — Re- 


S—Do- 
2 — Ti 


A—La- 

7—Si- 


6—Fa- 

\-5—Mi- 

4 — R&- 

3 — Do 
2 — Ti- 

k— La 


Fig.  10 


44 


MUSICOLOGY 


its  close  relationship  to  its  relative  major  (differing  by  only 
one  tone). 

2,  Minor  keys  have  no  signatures  of  their  own.  Relative 
minor  keys  use  the  signatures  of  their  relative  majors.  They 
are  recognized,  however,  by  the  accidental  sharp  or  natural  on 
5  of  the  major  key;  also,  when  the  music  begins  in  a  minor 
key,  the  bass,  and  sometimes  the  soprano,  begins  on  La  in- 
stead of  Do. 

3.  Observe  (Fig.  10)  that,  as  the  syllables  all  match  (ex- 
cept that  Sol  being  sharped  is  changed  to  Si),  there  is  no 
difference  in  reading  the  music  (with  the  syllables)  except 
that  the  accidental  on  5  of  the  major  (or  7  of  the  minor)  is 

called  Si.  Notice  that  the  5th  of  the 
major  is  sharped  and  becomes  the  7th  of 
the  minor;  so  that  the  rule  that  the  last 
sharp  is  ahvays  on  the  yth  of  the  key  holds 
good  in  minor  as  well  as  in  major  keys. 

4.  The  minor  key  (like  the  major) 
takes  the  name  of  the  letter  on  which  its 
key-note  (i)  is  placed,  and  which  is  always 
the  third  letter  below  the  key  letter  of  its 
relative  major.  Thus  the  relative  minor 
of  C  major  is  A  minor. 

5.  Set  the  minor  key  pattern  (Chart  I.) 
to  any  key  and  set  the  major  key  pattern 
to  the  corresponding  relative  major  key, 
and  compare. 

6.  In  Fig.  II  the  minor  key  pattern  is 
placed  so  that  its  key-note  is  opposite  the 
key-note  of  the  major  pattern.  It  Avill 
be  seen   that   the  figures  match  except  3 

and  6,  and  that  flatting  3  and  6  of  the  major  will  give  the 
minor  key  pattern.  This  is  called  the  tonie  minor  (the  key- 
note of  any  key  is  called  the  tonic).  The  tonie  minor  is  so 
called  because  its  tonic  corresponds  to  the  tonic  of  the  major, 


[J~"^?] 

r-1— Do-, 

-1—Ti- 

<i—La- 
-5—Sol- 

6— ie- 
~^Sol' 

•4 — Fa- 
-Z—Mi- 

-2—Re- 

i—Fa- 

3—3/6 
2— Be- 

-] — Do- 
^f—Ti- 

-1— Do- 
1-Ti- 

6—La- 

-5~Sol- 

6— ie- 

-l—Sol- 

■4—Fa^ 
-S—Mi- 

-2—Re- 

i—Fa.- 
2— i?e- 

-1 — Do- 
7—Ti 

1 — Do- 
1—Ti- 

Q~La- 
^—Sol 

6— ie 
^~Sol 

A— Fa 
-3— J/t- 

-2— i?e- 

i—Fa- 

-Z—Me- 
-2~Re- 

-1 — Do- 

-1—1)0- 

Fh:.  n 


ELEMENTARY  45 

and  from  this  view  is  more  closely  related  than  the  relative 
minor;  but  on  the  other  hand,  the  relative  minor,  differing 
from  the  major  by  one  tone,  is  more  closely  related  than 
the  tonic  minor,  which  differs  by  two  tones,  so  that  they  about 
balance  in  point  of  relationship  to  the  major. 

7.  The  toiiie  minor  of  any  major  key  is  also  the  relative 
minor  of  the  major  key  on  the  third  letter  above;  also,  the 
relative  minor  of  any  major  key  is  also  the  tonic  minor  of  the 
major  key  on  the  third  letter  below  (or  on  the  same  letter  as 
the  minor). 

8.  In  the  tonic  minor  (Fig.  11)  the  syllables  remain  as  in 
the  major,  except  that  JSIi  and  La,  being  flatted,  are  changed 
to  Me  and  Le  (see  Fig.  2,  p.  21,  and  observe  the  Italian  pro- 
nunciation). 

9.  Set  the  minor  key  pattern  (Chart  I.)  to  the  relative 
minor  of  C  major  by  setting  Do  (3)  opposite  C,  and  i  (La) 
opposite  A.  This  will  be  the  key  of  A  minor.  Now  set  the 
major  key  pattern  to  the  key  of  C  (relative  major  of  A  minor), 
and  observe  that  the  minor  is  formed  by  sharping  G,  which  is 
the  5th  of  the  major  or  the  7th  of  the  minor. 

10.  Now  set  the  major  pattern  to  the  key  of  A  (tonic 
major),  and  observe  that  the  minor  is  formed  by  making  nat- 
ural (equivalent  to  flatting)  C  and  F,  which  are  the  3d  and 
6th  of  both  the  major  and  minor  keys. 

11.  In  the  same  way,  set  the  minor  pattern  so  as  to  give 
the  relative  minor  of  all  the  sharp  keys  in  succession,  then 
all  the  flat  keys  in  succession  ;  and  at  each  move  comparing 
as  before  with  the  corresponding  relative  and  tonic  majors. 
Remember  that  removing  a  sharp  is  equivalent  to  flatting, 
and  removing  a  flat  is  equivalent  to  sharping. 

12.  The  Com.bined  Key  Table  on  next  page  shows  both 
the  major  and  minor  keys.  For  the  sake  of  distinction.  A, 
B,  C,  etc.,  represent  major,  and  a,  b,  e,  etc.,  minor  keys. 
Each  minor  key  is  represented  as  suspended  from  both  its 
relative  major  and  tonic  major.      (Evidently,  the  minor  keys 


46 


MUSICOLOGY 


12        3        4 
G       D       A        E 

$3 $0 fi- 

2  5  1 

1 4 $7- 

4  t"         $3         $6 

3 6 2 5- 

2  5  14 


5  6  7 

B  F«      C« 

-J3 Jfl $4- 

-Jt3 $G 3f2- 

«2      j5      jn 

-1 ^4 $7- 


COMBINKl)    MA.JOK    AND    MINOR  KKY  TABLE, 


ELEMENTARY 


47 


might  have  been  placed  between  the  major  keys  on  the 
same  staff,  but  this  would  have  been  more  confusing;  how- 
ever, staffs  having  the  same  clef  are  to  be  regarded  as  the 
same.) 

13.  Observe  that  each  minor  key  is  formed  either  by  sharp- 
ing 5  of  its  relative  major  or  by  flatting  3  and  6  of  its  tonic 
major,  and  that  therefore  the  relative  minor  key  and  tonic 
minor  key  do  not  mean  two  separate  keys,  but  one  and  the 
same  key ;  its  name  depending  on  which  major  key  signature 
it  is  used  with.  If  it  is  used  with  signature  of  its  relative 
major,  it  becomes  relative  minor  and  requires  an  accidental 
sharp  or  natural  on  5  of  the  major.  If  it  is  used  with  the 
signature  of  its  tonic  major,  it  becomes  tonic  minor  and  re- 
quires accidental  flats  or  naturals  on  3  and  6. 

14.  When  a  piece  of  music  is  written  in  or  begins  with  a 
minor  key,  the  signature  of  the  relative  major  is  used,  thus 
involving  the  fewest  accidentals.  This  is  also  more  conven- 
ient in  reading  music,  as  the  syllables  correspond  (except  Sol- 
Si). 

15.  In  each  of  the  three  sharp  keys  (B,  Y%,  C||)  on  the 
right  of  the  major  key  table,  5  of  the  key  is  already  sharped, 
therefore  to  form  the  relative  minor  of  each  of  these  keys  we 
would  have  to  double  sharp  the  5th  of  the  key.  Also  in  the 
three  flat  keys  (C  K  GK  D  b)  on  the  left,  where  3  and  6  are 
already  flatted,  would  require  double  flats  in  forming  their 
tonic  minors.  But  the  three  sharp  keys  on  the  right  and  the 
three  flat  keys  on  the  left  are  equivalent  or  interchangeable 
keys,  which  overlap  in  the  Key  Circle  (Fig.  9).  Therefore 
the  minor  key  table  completes  the  circle  of  minor  keys,  and 
any  extension  to  the  right  or  left  would  form  overlapping 
keys. 

16.  In  analyzing  the  minor  key  table,  to  study  the  relations 
existing  between  the  minor  keys,  we  must  take  into  account 
its  twofold  nature — tonic  and  relative. 

17.  Analyzing  the  table  from  the  tonic  view,  we  observe 


48  musicolo(;y 

that  each  minor  key  corresponds  to  its  tonic  major  as  to  the 
position  of  the  figures  on  the  staff.  Now  beginning  at  the 
left,  notice  that  4  of  each  key  is  made  natural  or  sharped  and 
becomes  7  of  the  next ;  or  beginning  at  the  right  and  revers- 
ing, 7  of  each  key  is  made  natural  or  flatted  and  becomes  4  of 
the  next.  This  relation  is  the  same  as  already  observed  in 
the  major  key  table  (4  and  7  being  same  in  both  tonic  major 
and  minor  keys). 

18.  Analyzing  the  table  from  the  relative  view,  we  observe 
that  6  and  2  of  any  minor  key  are  the  same  as  4  and  7  of  its 
relative  major  (see  Fig.  10,  p.  43).  Now,  beginning  at  the 
left,  we  notice  that  6  of  each  key  is  made  natural  or  sharped 
and  becomes  2  of  the  next ;  or  beginning  at  the  right  and  re- 
versing, 2  of  each  key  is  made  natural  or  flatted  and  becomes 
6  of  the  next.  This  is  also  the  same  relation  as  observed  in 
the  major  key  table,  but  from  the  relative  view. 

19.  We  observe  in  the  major  key  table,  beginning  at  the 
left,  that  4  made  natural  becomes  7  of  the  next  key,  and  re- 
mains natural  through  seven  keys,  then  sharped  and  remains 
sharped  to  end  of  table ;  or  beginning  at  the  right  and 
reversing,  7  made  natural  becomes  4  of  next  key,  and  remains 
natural  through  seven  keys,  then  flatted  and  remains  flatted 
to  end  of  table. 

20.  In  the  minor  key  table,  from  tonic  view,  beginning  at 
the  left,  we  observe  that  4  made  natural  becomes  7  of  next 
key,  and  remains  natural  (except  t?  3  and  1^6)  through  seven 
keys,  then  sharped  and  remains  sharped  (except  3  and  6)  to 
end  of  tabic;  or  beginning  at  the  right  and  reversing,  7  made 
natural  becomes  4  of  next  key,  and  remains  natural  (except 
b6  and  ^3)  through  seven  keys,  then  flatted  and  remains 
flattcil  to  end  of  table.  We  notice  that  6  and  3  (flat  or  nat- 
ural) form  exceptions  to  the  regular  order  shown  in  the  major 
key  table,  thus  indicating  their  accidental  character;  these 
being  the  tones  which  are  made  natural  or  flatted  to  form  the 
tonic  minor. 


ELEMENTARY  49 

2  1.  Again,  from  the  relative  view  (6  and  2  being  same  as  4 
and  7  of  relative  major),  beginning  at  the  left,  observe  that  6 
made  natural  becomes  2  of  next  key,  and  remains  natural 
(except  1^7)  through  seven  keys,  then  sharped  and  remains 
sharped  to  end  of  table ;  or  beginning  at  the  right  and  re- 
versing, 2  made  natural  becomes  6  of  next  key,  and  remains 
natural  (except  ^  7)  through  seven  keys,  then  flatted  and  re- 
mains flatted  (except  7)  to  end  of  table.  We  notice  that  7 
(sharp  or  natural)  forms  an  exception  to  the  regular  order, 
thus  indicating  its  accidental  character;  it  being  the  tone 
which  is  made  natural  or  sharped  to  form  the  relative 
minor. 

22.  Observe  also  that  these  three  figures  are  always  found 
side  by  side  in  the  same  order — thus  7,  3,  6 — and  that  from 
the  tonic  view  6  and  3  are  accidental  (7  being  regular  as  in 
tonic  major),  but  from  the  relative  view  7  is  accidental  (6  and 
3  being  regular  as  in  relative  major). 

23.  Now  combining  the  tonic  and  relative  natures,  begin- 
ning at  the  left  (regarding  7  as  accidental),  each  key  is  formed 
(toward  the  right)  by  making  natural  or  sharping  4  and  6  of 
one  key,  which  becomes  7  and  2  of  the  next ;  or  beginning  at 
the  right  and  reversing  (regarding  6  and  3  as  accidental),  each 
key  is  formed  (toward  the  left)  by  making  natural  or  flatting 
7  and  2  of  one  key,  which  become  4  and  6  of  the  next. 

24.  In  the  major  key  table  the  key  of  C  is  the  limit  of  both 
sharps  and  flats,  while  its  tonic  minor  is  the  limit  of  sharps 
and  its  relative  minor  the  limit  of  flats  in  the  minor  key 
table. 

25.  The  keys  of  ^^  and  (^/ minor  (where  the  sharp  and  flat 
sides  of  the  minor  table  overlap)  have  both  sharps  and  flats. 
These  keys  (being  between  the  relative  and  the  tonic 
minor  of  C  major)  may  be  called  the  iiitcrininor  keys  of 
C  major. 

26.  The  keys  of  rt' and  c  minor,  differing  from  the  key  of 
C  major  by  the  same  number  of  accidentals,  are  in  this  sense 


50  MUSICOLOGY 

equal  in  point  of  relationship  to  C  major;  however,  C  major 
is  more  closely  related,  to  its  tonic  (c)  minor  by  having  the 
same  key-note,  or  tonic.  In  the  key  of  ^/ minor,  a  flat  '(aE} 
used  as  a  flat)  is  understood  on  3  (3  being  sharp  in  its  tonic, 
D,  major). 

27.  The  minor  key  mode,  from  the  tonic  view,  is  best  seen 
in  the  key  of  c  minor,  as  it  contains  only  the  accidental  flats 
on  3  and  6  (there  being  no  signature  flats  in  its  tonic,  C, 
major),  and,  from  the  relative  view,  is  best  seen  in  the  key  of 
a  minor,  as  it  contains  only  the  accidental  sharp  on  7  (there 
being  no  signature  sharps  in  its  relative,  C,  major). 

28.  The  combined  tonic  and  relative  views  of  the  minor 
key  mode  is  best  seen  in  the  key  of  ^  minor;  the  tonic  minor 
being  seen  in  the  flats  on  3  and  6  (the  sharp  on  7  being  the 
signature  of  its  tonic,  G,  major),  and  the  relative  minor  being 
seen  in  the  sharp  on  7  (the  flats  on  3  and  6  being  the  signa- 
ture of  its  relative,  B  i^,  major). 

29.  Though  the  minor  mode  exists  in  the  other  minor  keys 
(all  conforming  to  the  same  pattern),  yet  it  is  not  so  plainly 
pictured  in  the  table,  owing  to  the  signature  sharps  or  flats 
(also  naturals,  understood  in  the  table)  involved. 

30.  In  the  sJiarp  minor  keys  the  minor  mode  is  seen  from 
the  relative  view  in  the  sharp  on  7  (the  other  sharps  belong- 
ing to  the  relative  major),  and  seen  from  the  tonic  view  in 
the  flat  naturals  (understood)  on  3  and  6  (3  and  6  being  sharp 
in  the  tonic  major). 

31.  In  \\\(t  fiat  minor  keys  the  minor  mode  is  seen  from  the 
relative  view  in  the  sharp  natural  (understood)  on  7  (7  being 
flat  in  the  relative  major),  and  seen  from  the  tonic  view  in 
the  flats  on  3  and  6  (the  other  flats  belonging  to  the  tonic 
major).  (Naturals  arc  expressed  in  music,  but  for  the  sake 
of  simplicity  arc  omitted  in  the  key  table.) 

32.  We  observe  that  in  minor  keys  the  last  flat  is  on  6 
(the  flat  on  3  being  the  same  as  the  flat  on  6  of  the  preced- 
ing key) ;   also,  that  the  flat  on  6  is  the  same  as  the  last  flat 


ELEMENTARY  5 1 

on  4  of  the   relative   major,  and  the   flat  on  3  is  the  same  as 
the  next  to  the  last  flat  on  i  of  the  relative  major. 

T,^.  Hence  we  have  the  general  rule:  T/ie  last  sJiarp  is 
akvays  on  the  ytJi  of  the  key  {major  or  minor);  the  last  fiat 
is  alivays  on  the  ph  in  major  keys,  and  on  the  6th  in  minor 
keys. 

34.  Relative  minor  keys  are  recognized  by  the  accidental 
sharp  or  natural  on  5  of  the  major  (7  of  minor).  Tonic  minor 
keys  are  recognized  by  the  accidental  flat  or  natural  on  6  or 
3,  or  both. 

35.  The  sharp  or  natural  on  7  is  for  the  purpose  of  form- 
ing the  half-step  between  7  and  i  (see  Fig.  10).  The  flats 
or  naturals  on  6  and  3  are  to  form  the  half-steps  between  5 
and  6  and  between  2  and  3  (see  Fig,   11). 

TjG.  In  major  keys  the  sharps  or  flats  of  any  key  are  retained 
in  the  following  key,  so  that  the  signature  of  each  key  in 
succession  is  built  up,  as  it  were,  by  adding  the  new  sharp  or 
flat  to  the  signature  of  the  preceding  key. 

T)"/.  In  relative  minor  keys  the  last  sharp  (on  7)  is  not  re- 
tained in  the  two  following  keys  and  is  therefore  accidental, 
so  that  it  could  have  no  place  in  building  the  signature  (the 
signature  of  the  relative  major  being  used  with  accidental 
sharp  on  7  of  the  minor,  or  5  of  major). 

38.  In  the  tonic  minor  keys  the  last  flat  on  6  is  retained  on 
3  of  the  following  key  but  is  dropped  in  the  next  key,  and  is 
therefore  accidental,  so  that  it  could  have  no  place  in  build- 
ing the  signature  (the  signature' of  the  tonic  major  being  used 
with  accidental  flats  on  6  and  3). 

39.  Bear  in  mind  that  the  sharps  and  flats  in  either  table 
always  show  what  letters  of  the  staff  are  to  be  sharped  or 
flatted,  and  that  the  figures  only  show  the  key  number  of  each 
letter  in  each  separate  key.  The  letters,  however,  are  only  the 
names  of  the  lines  and  spaces  of  the  staff,  so  that,  strictly 
speaking,  it  is  the  lines  and  spaces  (or  the  tones  they  repre- 
sent) that  are  sharped  or  flatted. 


52  MUSICOLOGY 

40.  Fii^".  9  (p.  42)  shows  the  major  key  table  bent  around 
in  a  circle.  In  the  same  way,  the  combined  key  table  maybe 
bent  around  in  a  circle,  forming  the  combined  major  and 
minor  key  circles  (the  minor  key  circle  being  complete  but 
not  overlapping). 

41.  Roth  major  and  minor  key  tables  may  be  extended, 
either  to  the  right  or  left  or  both,  by  merely  continuing  the 
same  relations.  Extending  the  table  in  either  direction  will 
involve  double  sharps  or  flats  (as  the  case  may  be).  The 
double  sharps  and  flats  will  occur  in  exactly  the  same  order 
as  the  first  sharps  and  flats,  and  the  key  letters  will  occur  in 
exactly  the  same  order  as  they  first  occur  (the  second  occur- 
rence of  any  key  letter  being  marked  |f  or  ^  ,  as  the  case  may 
be).  Of  course  extending  either  table  only  means  additional 
overlapping  keys  in  the  key  circle. 

42.  When  a  double  sharp  or  flat  occurs  in  a  piece  of  music 
it  is  as  an  accidental  on  a  line  or  space  already  sharped  or 
flatted  in  the  signature.  A  double  sharp  is  usually  written 
thus,  >jc-;  and  a  double  flat  thus,  !?).  When  thus  written  on  a 
line  or  space  already  sharped  or  flatted  in  the  signature,  it 
necessarily  includes  the  sharp  or  flat  on  the  same  line  or  space 
in  the  signature,  as  otherwise  the  note  affected  would  be 
triple  sharped  or  flatted. 

43.  A  double  sharp  raises  the  tone  a  whole  step.  A  double 
flat  lowers  the  tone  a  whole  step. 

44.  Keys  which  involve  an  extension  of  either  table  are 
used  only  in  modulation,  and  will  be  referred  to  again  under 
"  Modulation." 

MELODIC  MINOR  SCALE 

1.  The  key-note  (i)  of  any  key  is  the  controlling  tone  of 
that  key,  as  it  is  the  tone  of  complete  repose,  or  home  feel- 
ing, and  therefore  the  point  of  reference  with  which  each  tone 
is  compared  in  its  tonalit}'. 

2.  The  ton<"  a  half-step  below  the  key-note  is   called   the 


ELEMENTARY 


53 


leading  tone,  because  it  tends  to  lead   to  the  key-note,  there- 
by emphasizing  the  importance  of  the  key-note. 

3.  This  leading  tone  is  now  generally  regarded  as  an  essen- 
tial feature  of  all  keys,  major  or  minor,  in  order  to  give  due 
prominence  to  the  key-note  (the  trained  ear  seeming  to  re- 
quire it).  This  is  the  reason  for  sharping  the  7th  in  relative 
minor  keys,  as  otherwise  the  minor  key  would  have  no  lead- 
ing tone  a  half-step  below  the  key-note. 

4.  Sharping  7  leaves  an  augmented  interval  between  6  and 
7.      This  can   only  be  remedied  by  sharping  6  also,  thus: 


fc==j^fg^ 


•wi 


This   would  give  the  scale  too  much  of  a  major  character, 
as  will  be  seen  later. 

5.  As  a  leading  note  is  not  needed  descending  the  scale, 
we  may  (after  sharping  both  6  and  7  ascending  the  scale)  make 
both  natural  descending,  thus: 


This  is  called  the  Melodic  Minor  Scale,  but  cannot  cor- 
rectly be  called  a  key  (family  of  related  tones),  as  it  is  one 
family  ascending  and  another  family  descending;  nor  a 
mode,  as  it  is  one  mode  of  arrangement  ascending  and  an- 
other mode  descending.  In  minor  keys  the  ^6  and  tl  7  are 
only  used  as  melodic  or  passing  notes  (p.  87),  but  are  never 
used  in  chord  formations  as  harmony  notes. 

6.  Harmonic  Minor  is  the  general  name  for  both  relative 
and  tonic  minor,  to  distinguish  from  Melodic  Minor. 

THE  OLD  MINOR  MODE 

I.  The  original  or  old  viijior  pattern,  or  mode,  differs  from 
the  major  pattern  only  in  the  key-note  being  La  instead  of 
Do.      It  is  sometimes  found  in  old  tunes,  as  the  following: 


54 


MUSICULOGY 


Idumea. 


Observe  that  the  music  begins  and  ends  on  La,  and  that 
the  /th  (Sol)  is  not  sharped. 

2.  If  in  the  minor  key  table  (p.  46)  the  figure  7  is  made 
natural  where  sharped  and  flatted  where  natural,  the  table 
would  show  all  the  minor  keys  according  to  the  old  viinor 
mode.  Observe  that  each  key  would  differ  from  its  relative 
major  only  in  the  position  of  the  figures  on  the  staff,  and 
from  its  tonic  major  by  flats  or  naturals  on  3,  6,  and  7. 

3.  The  modern  minor  mode  (which  is  the  old  minor  Avith 
the  7th  sharped)  is  most  generally  adojitcd,  yet  the  old  minor 
mode  is  still  occasionally  used  ;  while  the  melodic  viinor  scale 
is  a  sort  of  compromise. 

THE  ANCIENT  GREEK  MODES 
I.    Observe   that  in  the  major  pattern   each  octave  is  made 
up  of  five  whole  steps  and  two  half-steps,  the  mode  of  arrange- 
ment placing   the   half-steps  between    3   and  4,  and   7  and  i  ; 
I     z    i    4    5     (5    7      bi-it  the  steps  and  half-steps  may   be  ar- 
ranged in  seven  principal  ways,  as  shown 
in   h^ig.    13.      These  were  all  used  in  the 
music  of  the  Ancient  Greeks,  and  hence 
are  called  the  Ancient  Greek  Modes. 

2.  After  the  science  of  Harmony 
(combining  of  tones)  began  to  develop, 
it  was  found  that  two  of  these,  the  1st 
and  tlie  6th  (corresponding  to  our  major 
and  old  minor  modes),  were  best  suited 
''"■■  '^-  for  Jiarmoni;:ation.      The  others,   there- 

fore, have  become  obsolete. 


El.E.MEMAKV  55 

3.  Observe  that  the  word  iiwdc  refers  to  the  mode  of  arran<re- 
ment  of  the  steps  and  half-steps,  and  the  word  key  refers  to 
the  mode  as  adjusted  to  a  certain  pitch.  The  word  pattern 
refers  to  the  means  of  representing  the  intervals  of  the  mode  to 
the  eye,  thus  being  to  the  eye  what  the  mode  is  to  the  ear. 

4.  We  see  that  while  there  are  only  two  modes  now  in  com- 
mon use,  yet  by  adjusting  the  key-note  of  each  of  these  modes 
successively  to  correspond  in  pitch  to  the  twelve  different 
half-steps  into  which  the  octave  is  divided,  we  get  twelve 
major  and  twelve  minor  keys, 

THE  HARMONIC  SCALE  NAMES 

I.  The  1st  of  any  key  is  called  the  Tonic,  or  key-note;  it 
is  the  tone  upon  which  the  key  is  founded,  and  is  the  point 
of  repose  or  home  feeling,  and  the  point  from  which  all  the 
other  tones  of  the  key  are  measured.  The  2d  of  any  key  is 
called  the  Super-tonic  (meaning  above  the  tonic) ;  the  jd  of 
any  key  is  called  the  A/ediant  (midway  between  the  tonic  and 
the  dominant) ;  the  ^t/i  of  any  key  is  called  the  Siib-doviinant 
(dominant  below  the  tonic — being  the  5th  below  the  tonic 
above);  the  f/'//  of  any  key  is  called  the  Dominant  (domina- 
ting over  the  other  tones  in  the  closeness  of  its  relation  to 
the  tonic) ;  the  6tJi  of  any  key  is  called  the  Sub-nicdiant 
(mediant  below  the  tonic — midway  betw^een  the  sub-domi- 
nant and  tonic  above) ;  the  "jtJi  of  any  key  is  called  the 
Sub-tonic  (below  the  tonic) — it  is  also  called  the  leading 
tone,  as  it  leads  to  the  tonic  above.  Thus  the  super- 
tonic,  mediant,  and  dominant  (being  more  closely  related 
to  the  tonic  below)  are  regarded  as  above  the  tonic, 
and  are  together  called  the  dominant  side  of  the  scale  ;  while 
the  sub-tonic,  sub  mediant,  and  sub-dominant  (being  more 
closely  related  to  the  tonic  above)  are  regarded  as  below  the 
tonic  {sub  meaning  belozv),  and  are  together  called  the  sub- 
dominant  side  of  the  scale.  The  tonic  is  thus  the  center,  or 
pivot,  around  which  the  key  forms. 


$6  MUSICOLOGY 

INTERVALS 

1.  Intervals  are  the  distances  between  sounds  as  regards 
pitch.  The  interval  from  i  to  2  is  called  a  second;  i  to  3,  a 
third;  i  to  4,  a  fourth;  i  to  5,  a  fifth;  i  to  6,  a  sixth;  i 
to  7,  a  seventh  ;    i  to  i  above,  an  octave. 

2.  The  natural  intervals  upward  from  i  of  the  major  key 
pattern  are  taken  as  the  standards  of  measurement  and  called 
major,  except  the  ist,  4th,  5th,  and  octave,  which  are  called 
perfect,  thus:  i  to  i  (unison)  is  called  a  perfect  ist,  or  prime; 
I  to  2,  equal  to  i  whole  step  (see  major  pattern),  is  called  a 
major  2d;  i  to  3,  equal  to  2  whole  steps,  is  called  a  major 
3d;  I  to  4,  equal  to  2^  steps,  is  called  a  perfect  4th;  i  to  5, 
equal  to  3  J  steps,  is  called  a  perfect  5th  ;  i  to  6,  equal  to  4^ 
steps,  is  called  a  major  6th;  i  to  7,  equal  to  5L  steps,  is 
called  a  major  7th  ;  i  to  1  (above),  equal  to  5  steps  and  two 
I  steps,  is  called  a  perfect  octave.  A  9th  (compound  2d), 
loth  (compound  3d),  etc.,  are  the  same  intervals  extended 
beyond  the  compass  of  an  octave  and  are  called  compound 
intervals,  while  the  intervals  within  the  compass  of  an  octave 
are  called  simple. 

3.  A  minor  interval  is  J  step  less  than  major;  a  diminished 
interval  is  l  step  less  than  minor  or  the  perfects  (ist,  4th,  5th, 
8ve) ;  an  augmented  interval  is  J  step  greater  than  major  or 
the  perfects, 

4.  With  these  standards  we  can  measure  the  interval  be- 
tween any  two  tones  of  the  scale. 

5-  Now  analyze  all  the  intervals  of  the  major  key  pat- 
tern, beginning  with  2  instead  of  i  ;  then  beginning  with  3, 
etc.  Analyze  all  the  intervals  of  the  minor  key  pattern  in  the 
same  way. 

6.  Invertetl  intervals  are  those  with  the  tones  inverted: 
thus,  I  to  2  inverted  is  2  to  i  above,  etc.  Therefore  a  2d  in- 
verted becomes  a  7th,  a  3d  becomes  a  6th,  etc.  Also  a  dim- 
inished interval  inverted  becomes  an  augmented  interval,  and 
vice  versa;  a  minor  interval   inverted   becomes  a  major,  and 


ELEMENTARY  57 

vice  versa.  The  perfect  intervals  inverted  remain  perfect  in- 
tervals:  thus,  a  perfect  ist  inverted  becomes  a  perfect  octave, 
and  vice  versa;  a  perfect  4th  inverted  becomes  a  perfect  5th, 
and  vice  versa.  This  is  because  i  and  i  are  the  extremes  of  the 
octave,  while  4  and  5  together  are  the  middle  of  the  octave. 

7.  The  natural,  or  diatonic,  intervals  of  any  key  are  those 
which  do  not  involve  accidentals.  Those  involving  acci- 
dentals are  called  chromatic  or  altered  intervals. 

8.  Observe  that  in  figuring  intervals  we  count  both  limits : 
thus,  5  to  7  is  a  3d,  because  5  and  7  are  both  counted  (involv- 
ing three  figures — 5,  6,  7). 

9.  I  to  I  H  and  I  to  2  I?  are  evidently  the  same  interval  (J 
step),  but  differently  figured  ;  the  first  involves  but  one  fig- 
ure, or  degree,  and  is  an  augmented  ist,  while  the  second  in- 
volves two  figures  and  is  a  minor  2d. 

10.  Observe  by  comparing  that  the  following  intervals  are 
equal:  augmented  ist  =  minor  2d  (each  \  step);  major  2d  = 
diminished  3d  (i  step);  aug.  2d  =  minor  3d  (U-  steps); 
major  3d  =  dim.  4th  (2  steps);  aug.  3d  =  perfect  4th  {2\ 
steps);  aug.  4th  =  dim.  5th  (3  steps);  perfect  5th  =  dim. 
6th  (3I  steps);  aug.  5th  =  minor  6th  (4  steps);  major  6th=: 
dim.  7th  (4I  steps);  aug.  6th  =  minor  7th  (5  steps);  major 
7th  =  dim.  octave  (5|  steps);  aug.  7th  =  octave  (6  steps). 
Thus  by  an  enharmonic  change  (p.  41:  13)  of  one  tone, 
each  interval  may  be  expressed  in  two  ways. 

11.  The  major,  minor,  and  perfect  intervals  are  most  used. 
The  intervals  of  less  frequent  use  are  augmented  2ds,  4ths, 
5ths,  and  6ths,  and  diminished  5ths  and  7ths.  All  these,  ex- 
cept the  aug.  6th,  are  to  be  found  as  diatonic  intervals.  The 
aug.  2d  is  found  between  6  and  7  of  minor  keys  (see  pat- 
tern) ;  the  aug.  4th,  between  4  and  7  in  major  and  minor 
keys;  the  aug.  5th,  between  3  and  7  in  minor  keys;  the  dim. 
5th,  between  7  and  4  in  major  keys  and  between  7  and  4, 
and  2  and  6,  in  minor  keys;  and  the  dim.  7th,  between  7 
and  6  in  minor  keys.      Otherwise  they  are  chromatic  intervals. 


58  MUSICOLOGV 

Intervals    may    be    chromatic    in    one    key   and    diatonic    in 
another. 

12.  Intervals  are  also  concordant  (smooth)  or  discordant 
(rough).  The  perfect  concords  are  the  octave,  perfect  5th, 
and  perfect  4th.  The  imperfect  concords  are  the  major  and 
minor  3ds  and  6ths.  The  discords  are  the  major  and  minor 
2ds  and  Jths,  and  augmented  and  diminished  intervals. 

13.  Observe  that  intervals  are  reckoned  upward  (not  down- 
ward, unless  so  stated),  because  the  lowest  note  is  the  bass 
(base),  or  foundation ;  and  calculations  are  therefore  made 
from   it  instead  of  from  the  highest  note. 

14.  Nature  divides  the  music  scale  into  octaves,  and  sub- 
divides the  octaves  into  perfect  5ths,  plus  perfect  4ths.  These 
are  called  the  perfect  intervals.  Taking  the  difference  be- 
tween the  perfect  5th  and  perfect  4th  we  get  the  interval  of  a 
whole  step.  Taking  this  as  a  measure,  we  find  that  it  is  con- 
tained in  the  perfect  4th  two  and  one-half  times,  and  in  the 
perfect  5th  three  and  one-half  times.  Beginning  at  i  and 
measuring  upward,  the  half-steps  come  between  3  and  4,  and 
between  7  and  i,  thus  forming  the  major  diatonic  's,zd\t.  Now 
using  the  half-steps  as  a  measure,  the  whole  steps  will  be 
divided  into  half-steps,  thus  forming  the  chromatic  scale. 

15.  To  assist  in  recognizing  the  intervals  on  the  staff  we 
may  observe  that  intervals  represented  by  the  odd  figures 
(3ds,  5ths,  7ths,  etc.)  are  like  situated  (both  notes  forming 
the  interval  being  on  lines  or  spaces),  and  the  intervals  rep- 
resented by  the  even  figures  (2ds,  4ths,  6ths,  8ves,  etc.)  are 
unlike  situated  (one  note  being  on  a  line  and  the  other  on  a 
space) ;  also  that  the  interval  between  the  bas^  and  treble 
staffs  counts  only  for  a  5th,  regardless  of  the  distance  of  sep- 
aration between  the  staffs. 


PART  SECOND 


STRUCTURE   OF    AlUSIC 


HARMONY 

1.  Harmony  relates  to  the  combining  of  tones.  When 
two  or  more  tones  that  are  sounded  at  the  same  time  blend, 
they  are  said  to  harmonize  or  chord.  If  they  do  not  blend, 
they  are  said  to  discord. 

2.  Tones  are  said  to  be  consonant  or  dissonant  according 
as  they  chord  or  discord.  Any  two  tones  that  form  between 
them  the  interval  of  a  2d  are  dissonant,  while  any  two  tones 
forming  between  them  the  interval  of  a  3d  are  consonant. 
Therefore  it  may  be  said  that  harmojiy  builds  in  jds. 

3.  The  alternate  tones  of  the  scale  chord,  while  the  consec- 
utive tones  discord.      Therefore   the  harmonic  scale  would  be 

Do,  Mi,  Sol,  fi,  Re,  Fa,  La,  Do. 

4.  The  Triad. — Any  three  alternate  tones,  as  3,  form  a 
complete  chord  ;  but  if  we  add  7  it  will  form  a  discord  and 
give  to  the  whole  chord  a  dissonant  character.  This  is  be- 
cause 7  forms  the  interval  of  a  2d  with  i  next  above  in  the 
scale,  and  i  is  already  a  member  of  the  chord.  (In  a  har- 
monic sense  all  tones  of  the  same  name,  though  in  different 
octaves,  are  regarded  as  the  same.) 

5.  Again,  if  we  take  the  chord  5  and  add  2,  it  will  discord 
with  3  next  above  in  the  scale,  and  which  is  already  a  mem- 
ber  of   the  chord.      For  the  same  reason    we   will    find   that 


6o  MUSICOLOGV 

every  chord  of  more  than  three  tones  will  be  dissonant.  There- 
fore the  only  perfect  chord  is  the  three-toned  chord. 

6.  The  three-toned  chord  is  called  the  couimon  chord,  or 
triad,  and  is  the  basis  of  all  harmony. 

Ri'iJiark. — Chords  expressed  by  the  scale  figures  (thus,  i^ 
etc.)  are  (like  the  key  pattern)  applicable  alike  to  all  the  keys; 
while  if  expressed  by  the  fixed  letters  of  the  staff  (thus,  e  ,etc.), 

they  would  be  expressed  by  different  letters  in  the  difTerent 
keys,  and  would  have  to  be  memorized  individually  in  all  the 
keys,  which  is  impracticable.  The  figures  are  movable  and 
represent  general  principles,  while  the  letters  are  local  and 
represent  fixed  tones.  The  figures  always  represent  the  let- 
ters, but  the  same  figures  represent  different  letters  in  differ- 
ent keys.  Therefore,  in  studying  the  general  principles  of 
chords  it  is  better  to  use  the  figures. 

7.  Four-part  harmony  embraces  four  parts,  or  voices:  bass, 
tenor,  alto,  and  soprano.  To  make  the  three-toned  chord  a 
four-voiced  chord,  one  of  the  tones  will  have  to  be  doubled. 
This  is  not  adding  a  new  tone,  but  simply  doubling  one  of 
the  tones  (same  tone  in  different  octave).  Though  the  triad 
thus  becomes  a  four-voiced  chord,  it  is  still  a  triad  (three- 
toned). 

8.  Any  tone  may  be  taken  as  the  root  or  foundation  of  a 
chord,  and  when  thus  used  it  is  called  the  root  of  the  chord. 
The  triad  is  then  built  by  adding  a  tone  a  3d  above,  then  an- 
other a  3d  above  that  (or  a  5th  above  the  root).  Hence  a 
triad  (common  chord)  consists  of  a  root  note  with  its  3d  and 
5th.  (The  second  and  third  members  of  a  triad  are  always 
referred  to  as  the  3d  and  5th,  thus  expressing  the  intervals 
of  the  triad;    as  ist,  3d,  5th — not  1st,  2d,  3d.) 

9.  In  doubling  one  of  the  tones  to  form  four-part  harmony, 
it  is  usually  best  to  double  the  root,  as  it  is  the  principal  tone  ; 
and  better  to  double  the  5th  than  the  3d.  If  the  3d  is  minor 
it  may  sometimes  be  doubled  ;    but  the   major  3d  should  rare- 


STRUCTURE    OF    MUSIC 


6i 


ly  be  doubled,  because  of  its  tendency  to  ascend  a  half-step; 
and  doubled  3ds,  being  an  octave  apart  and  moving  in  the 
same  direction,  will  produce  consecutive  octaves,  which  are 
forbidden  (see  p.  80  :ig).  On  the  other  hand,  the  3d  should 
rarely  be  omitted. 

10.  A  triad  takes  the  name  of  its  root  note  and  is  repre- 
sented, for  convenience,  by  the  corresponding  roman  numer- 
al, thus:  the  triad  of  the  Tonic,  I.  ;  the  triad  of  the  Super- 
tonic,  II.  ;  the  triad  of  the  Mediant,  III.  ;  the  triad  of  the 
Sub-dominant,  IV.  ;  the  triad  of  the  Dominant,  V.  ;  the 
triad  of  the  Sub-mediant,  VI.  ;  and  the  triad  of  the  Sub- 
tonic,  VII. 

11.  A  triad  (common  chord)  is  called  major  when  it  is 
made  up  of  a  major  3d  (2  steps)  and  perfect  5th  (3] 
steps)  above  the  root,  and  called  viiiior  when  made 
up  of  a  minor  3d  (li  steps)  and  perfect  5th  above 
the  root.  The  perfect  5th  is  made  up  of  a  major 
3d  and  a  minor  3d,  one  above  the  other;  in  the 
major  triad,  the  major  3d  is  below  and  the  minor  3d 
above  ;  in  the  minor  triad,  the  minor  3d  is  below  and 
the  major  3d  above. 

12.  A  diminislicd  triad  is  made  up  of  a  minor  3d 
and  diminished  5th  (or  two  minor  3ds  one  above  the 
other).  An  aiig}>icnted  triad  is  made  up  of  a  major 
3d  and  augmented  5th  (or  two  major  3ds,  one  above 
the  other). 

13.  Analyzing  the  triads  of  the  major  key  pattern, 
counting  the  steps  and  half-steps  in  each  interval,  we 
find  that  the  Tonic  triad  (  »  jj^  )  is  made  up  of  a  major 
3d  (2  steps)  andperfect  5th  (3i  steps),  and  is  therefore  a 
major  triad  ;  Super-tonic  triad  (  %  \^  ),  made  up  of  a  minor 
3d  (i  i  steps)  and  perfect  5th,  is  a  minor  triad  ;  Mediant 
triad  (5801  ),  made  up  of  a  minor  3d  and  perfect  5th,  is  a  minor 
triad;    Sub-dominant  triad  (|La),  made  up  of  a  major  3d  and 


-1 — Do 

7 — Ti-\ 

6 — La 

-5— Sol 

-i—Fa 
■Z—Mi- 

2 — Re- 

-1 — Do- 
7—Th 

6 — La 

5— Sol 

-'i—Re 

-1 — Do 

u — Th 

6 — La- 
■5— Sol 
A— Fa 

2— i?e- 
4 — Do- 


62 


MUSICOLOGV 


rl — La-i 

-Si- 


-Q—Fa 
5—Mi- 


-i—Ee- 


-La 


perfect  5th,  is  a  major  triad.  In  the  same  way,  we  find  that 
the  Dominant  triad(7£)  is  a  ;//<'?;'i^r  triad  ;  the  Sub-mediant 
(11;°)  is  a  vii)ior  triad;  and  the  Sub-tonic  ( s^f )  is  a 
diminished  triad  (made  up  of  a  minor  3d  and  dim- 
inished    5th  —  2|=3steps). 

14.  Analyzing  the  triads  of  the  minor  key  pat- 
tern in  the  same  way,  we  find  that  the  Tonic  triad 
(  3  »° )  is  a  minor  triad  ;  the  Super-tonic  (  f  i^  )  is  a  dim- 
inisJied  triad  ;  the  Mediant  (  g  |' )  is  an  augmented  triad 
(made  up  of  a  major  3d  and  augmented  5th — 4 
steps);  the  Sub-dominant  (|};^)  is  a  minor  triad; 
the  Dominant  ( |  ^|i )  is  a.  major  triad;  the  Sub-medi- 
ant (  J  ^° )  is  a  major  triad  ;  and  the  Sub-tonic  ( |  :jf )  is 
a  diminished  triad. 

15.  Representing     the     major      triads     by     large 
h—Mi-    roman  numerals,    the    minor  triads  by   small    roman 

numerals  ;    the  dinmiished  triads,  same  as  minor  with 
(°)  added  ;   and   the  augmented  triad,  same  as  major 
with    (')    added,    all  the    triads    of    both    major    and 
minor  keys  w^ould  be  represented  thus: 

Triads  of  major  keys — I,  11,  ni,  IV,  V,  vi,  vii'. 
Triads  of  minor  keys — i,  h°,  III',  iv,  V,  VI,  vu°. 
Or  analyzed  thus: 

I  V"  n  VI 


5 — Mi 


-i—Re 


3— Do- 
2 — Ti- 


■1 — La 

-7 — SH 


6—Fa- 
5—Mi- 

4 — Ee 

-3 — Do- 

-2 — Ti- 

-1 — ia-1 


Do      Mi      Sol     Ti       Re      Fa       La       Do     Mi 
Triads  of  major  keys — i 3...    5 7...    2...   4. . .  .C. . .    i 3 


IV 


VI 


La     Do      Mi      Si        Ti      Re      Fa      La     Do 
Triads  of  minor  keys — i...   3 5 7...   2...   4...   6 1...'3 


III'  vn°  IV 

Dots  show  number  of  half-steps  between;    . . . .  =  majnr  31I,  ...    =  minor  3<1. 


STRUCTURE    OF    MUSIC  63 

16.  We  see  that  major  keys  have  three  major  triads  (I,  IV, 
V),  three  minor  triads  (II,  III,  VI),  and  one  diminished  triad 
(Vll°);  and  that  minor  keys  have  two  minor  triads  (I,  iv), 
two  major  triads  (V,  VI),  two  diminished  triads  (ll°,  Vll°), 
and  one  augmented  triad  (HI'). 

17.  Major  triads  have  a  bold,  aggressive  character,  while 
minor  triads  have  a  subdued,  tempering,  plaintive  character. 

18.  We  have  seen  (p.  58:14)  that  the  octave,  perfect  4th, 
and  perfect  5th  are  the  principal  divisions  of  the  scale;  and 
evidently,  therefore,  the  divisional  points  i,  4,  5  are  the  prin- 
cipal points  of  the  scale.  We  noticed  also  the  prominence  of 
these  intervals  in  the  interrelationship  of  keys  as  shown  by 
the  major  key  table  (p.  40) ;  that  in  the  ascending  direction 
(left  to  right),  the  dominant  (5)  of  any  key  is  the  point  around 
which  the  next  key  forms  (called  therefore  the  key  of  the 
dominant);  and  in  the  descending  direction  (right  to  left), 
the  sub-dominant  (4)  of  any  key  is  the  point  around  which 
the  next  key  forms  (called,  therefore,  the  key  of  the  sub-dom- 
inant). In  the  first  instance,  raising  the  key-note  (i)  a  per- 
fect 5th  or  lowering  it  a  perfect  4th;  and  in  the  second  in- 
stance, raising  the  key-note  a  perfect  4th  or  lowering  it  a  per- 
fect 5  th. 

19.  Naturally,  therefore,  the  perfect  intervals  (octave,  4th, 
and  5th)  are  the  most  prominent  intervals,  and  the  tones  i, 
4,  5,  the  most  prominent  tones,  and  the  triads  I,  IV,  V  the 
most  prominent  triads,  of  a  key.  For  this  reason,  the  triads 
I,  IV,  V  are  called  the  characteristic  harmonies,  as  they  not 
only  determine  the  character  of  the  key  (whether  major  or 
minor),  but  also  form  the  background  or  framework  of  all 
music.  Any  tune,  or  melody,  can  be  harmonized  by  these 
three  chords  alone,  as  the  three  together  contain  all  the  tones 
of  the  key.  These  chords  (with  the  Dominant  7th)  are  the 
ones  commonly  used  in  extemporizing  accompaniments  on 
piano,  guitar,  or  other  instruments. 

20.  These   characteristic    chords    (I,    IV,    V)  are   the   only 


64  MUSICOLOGY 

major  triads  in  any  major  key,  and  determine  its  major  char- 
acter ;  I  and  IV  are  the  only  minor  triads  in  any  modcrti 
minor  key,  and  determine  its  minor  character.  In  the  old 
minor  mode  (p.  53),  all  three  characteristic  chords  (I,  IV,  V) 
are  minor,  thus  makin<^  it  more  distinctly  minor  (being  sym- 
metrically the  opposite  of  the  major  mode).  But  the  leading 
tone  (i  step  below  the  key-note)  is  now  generally  considered 
essential  in  order  to  give  due  prominence  to  the  key-note, 
thus  making  it  necessary  to  raise  the  7th  of  the  old  minor 
mode  a  half-step.  This  makes  the  dominant  triad  (V)  major. 
2  I.  Hence  the  law  :  TJic  Dominant  Chord  must  he  major  in 
both  major  and  minor  keys. 

22.  If  we  should  also  raise  the  6th  a  half-step  (see  p.  53: 
4),  to  avoid  the  augmented  interval  between  6  and  7,  then 
the  Sub-dominant  chord  (iv)  would  also  become  major,  and 
the  minor  key  (two  of  its  three  characteristic  triads  being 
major)  would  have  too  much  of  a  major  character. 

23.  Hence  the  law:  The  Tonic  (l)  and  the  Sub-dominant 
{\\')  triads  must  both  be  minor  in  minor  keys. 

24.  Since  these  laws  determine  the  form  of  the  character- 
istic triads  of  minor  keys,  they  necessarily  fix  the  mode,  or 
pattern,  of  minor  keys. 

POSITIONS    AND    FORMS    OF    THE    TRIAD 

1.  On  the  principle  that  harmony  builds  in  3ds,  the  natural 
or  direct  position  or  form  of  any  chord  is  that  in  which  its 
notes  are  arranged  in  3ds;  and  the  lowest  note  is  therefore 
regarded  as  the  root  or  foundation  of  the  chord.  The  chord, 
however,  may  be  taken  in  various  positions  and  forms,  but  as 
long  as  the  same  tones  are  used  (counting  octaves  as  merely 
replicates  of  the  same  tone)  it  is  regarded  as  the  same  chord. 

2.  The  triad  has  three  positions  with  reference  to  the  so- 
prano, or  highest  note,  thus:  if  the  root  of  the  triad  is  in  the 
soprano,  the  triad  is  in  1st  position  ;  if  the  3d  is  in  the  soprano, 
the  triad  is  in  2d  position  ;  if  the  5th  is  in  the  soprano,  the 
triad  is  in  3d  position. 


STRUCTURE    OF    MUSIC  65 

3.  The  triad  has  three  forms  with  reference  to  the  bass,  or 
lowest  note.  The  lowest  note  is  always  the  bass  (whether 
root,  3d,  or  5th).  The  bass  (base),  as  its  name  implies,  is 
the  base  or  support  of  the  entire  chord.  When  the  root  of 
tlie  triad  is  the  bass,  it  is  called  root  bass  and  the  triad  direct. 
If  the  root  of  the  triad  is  not  the  bass,  the  triad  is  said  to  be 
inverted.  If  the  3d  is  the  bass,  it  is  called  the  ist  inversion; 
if  the  5th  is  the  bass,  it  is  called  the  2d  inversion. 

4.  Thus  we  see  that  \.\\q  position  (ist,  2d,  3d)  is  determined 
by  the  soprano  or  highest  note,  and  \.\\q  forjii  (direct  or  in- 
verted) by  the  bass  or  lowest  note  (bass  and  soprano  being  the 
outside  and  therefore  most  prominent  parts).  A  triad  is  strong- 
est and  most  independent  when  in  ist  position  with  root 
bass,  the  root  of  the  triad  being  thus  in  both  bass  and  soprano. 

5.  The  treble  staff  has  to  do  mainly  with  position,  and  the 
bass  staff  with  form.  In  playing  the  piano  or  organ,  the 
right  hand  has  to  do  with  position  ;  the  left  hand,  with  form. 

6.  Position  and  form  are  entirely  independent  of  each 
other, so  that  any  form  of  the  triad  (root,  ist  inv.or2d  inv.)  may 
be  used  with  any  position  of  the  triad  (ist,  2d,  or  3d).  To 
determine  the  position,  we  need  look  only  at  the  highest  note; 
to  determine  the  form,  we  need  look  only  at  the  lowest  note. 

7.  The  different  positions  and  forms  of  chords  relieve  the 
monotony  which  would  result  from  using  only  one  form  and 
position  of  each  chord.  They  are  also  the  necessary  result 
of  voice  leading,  as  will  be  seen  later.  (To  avoid  the  incon- 
venience of  using  both  words,  we  may  use  either  of  the 
words — position  or  form — in  a  general  sense  ;  but  a  chord  is 
usually  referred  to  as  direct  or  inverted.) 

8.  Fig.  14  shows  the  different  positions  and  forms  of  the 
tonic  triad  in  the  key  of  C  major. 

9.  When  four-part  harmony  is  written  so  that  the  three 
upper  parts  are  within  the  compass  of  an  octave,  so  they  can 
be  played  with  one  hand,  it  is  called  close  harmony;  other- 
wise,   it    is   called    open   harmony.      In    open    harmonv,  tones 


66 


MUSICOLOGY 


proper  to  a  cliord  arc  omitted   between    two    or   more  of   the 
three  upper  parts. 


pos. 

ISl 

pos. 

2d 
pos. 

I  St 

pos. 

2d                3d 
pos.           pos. 

m 

2d 
pos. 

id              1st 
pos.          pos. 
1 

— ll 

-^ 

~~^ 

— r — K— 

.'  • 

\ M 

W'l  t— 

-i — 

-5 

—3 

— »■ — T — 

u_- 

— r — 3^- 

Close 

Harmony 

1 

-•- 

1 

Open  Harmony 

-0- 

-0-      • 

Open  Harmony 

1              J 

/^"\  •      ^        1 

J 

1 

L          # 

~  1* 

!           d       1 

i^J»      ^        j 

|i 

1*               m 

1             ^ 

V.     V         "i          m 

m 

« 

■ 

5 

. — 1 — 

1 f 

~ • •' 

6 

6               6 

6 

6 

6 

irect  Form 

ist  Inversion 
Fig.  14. 

4 

4 
2d  Inversion 

4 

10.  In  V'v^.  14  the  figures  on  the  staff  at  the  left  show  all 
the  tones  proper  to  the  tonic  triad  in  the  key  of  C 

1 1.  Close  harmony  is  generally  used  for  male  or  for  female 
voices  alone ;   open  harmony,  for  mixed  voices. 

12., Close  harmony  is  usually  best  adapted  for  loud,  full 
expression,  and  open  harmony  for  soft,  delicate  expression. 
As  a  general  rule,  however,  it  is  best  to  keep  the  parts  as 
nearly  as  possible  equidistant.  Not  more  than  an  octave 
should  ever  intervene  between  any  two  contiguous  parts, 
exc.ept  the  bass  and  tenor. 

THOROUGH-BASS    FIGURING   OF    TRIADS 

(Fipuriiig   from    the   bast-.) 

1.  In  the  direct  form  of  any  triad,  as  |  (taking  the  triad  I 
for  example),  the  intervals  figured  from  the  bass  are  ^'^. 
Therefore,  the  figures  ^  (representing  the  intervals  of  the 
triad)  stand  for  the  direct  form  of  the  triad.  When  no  fig- 
ures arc  expressed,  this  form  is  understood. 

2.  In  the  1st  inversion  (.5)  the  intervals  figured  from  the 
bass  are  :1,i\  Therefore,  the  figures  |,  or  simply  6  (3  being 
understood),  stand  for  the  ist  inverted  form  of  the  triad. 

3.  In  the  2d  inversion  (  ?  )  the  intervals  figured  from  the 
bass  are  5!h-  Therefore,  the  figures  |  stand  for  the  2d  inverted 
form  of  the  triad. 


STRUCTURE    OF    MUSIC  6^ 

4.  Again  (taking  the  triad  IV  for  example),  the  direct 
form  (root  bass)  would  be  \ — intervals-^:,*",  istinv.  |  —  inter- 
vals I?  ;  2d  inv.  4  —  intervals  lit.  All  triads  are  figured  the 
same  way,  therefore  the  figures  6  and  %  are  merely  to  show 
the  inversions  of  the  triad,  and  when  used,  are  usually  writ- 
ten under  the  triad  (see  Fig.  14).  Observe  that  the  triads 
are  figured  as  if  the  notes  were  all  in  the  same  octave  (octaves 
being  regarded  as  merely  replicates  of  the  same  tone). 

5.  The  key  figures  and  thorough-bass  figures  should  not  be 
confused,  as  they  do  not  correspond  except  where  the  chord 
intervals  and  the  key  intervals  happen  both  to  be  figured  from 
the  key-note;  as  the  \  of  the  I  triad,  the  \  of  the  IV  triad, 
and  the  %  of  the  VI  triad. 

6.  It  should  also  be  fixed  in  the  mind  that  thorough-bass 
figuring  is  merely  a  system  of  measurement  from  the  lowest 
note,  and  has  nothing  whatever  to  do  with  the  generic  nature 
of  a  chord.  Thorough-bass  figuring  is  useful  in  training  the 
mind  to  measure  all  intervals  from  the  bass. 

7.  As  a  matter  of  convenience,  we  may  speak  of  the  3  chord 
the  6  chord,  the  %  chord,  etc.,  bearing  in  mind  that  these  ex- 
pressions refer  to  the  form,  and  not  to  the  nature  of  the  chord. 

DISSONANT    CHORDS 

1.  Dissonant  chords  (or  discords)  are  not  always  unpleas- 
ant, but  often  quite  the  reverse  in  relieving  the  monotony  of 
too  much  concord.  They  make  the  concords  more  effective 
by  contrast,  and  increase  the  means  of  expression,  therefore 
they  are  very  important  in  music ;  however,  they  must  be 
short  and  always  resolve  into  their  proper  concords. 

2.  The  major  and  minor  triads  are  the  only  concords;  since 
in  their  different  positions,  while  containing  only  consonant 
intervals,  they  contain  all  the  possible  consonant  intervals. 

3.  The  Dissonant  Triads.  Diminished  and  augmented 
triads  are  dissonant,  because  they  contain  the  interval  of  a 
dissonant  5th — the  dim.  triad  containing  a  dim.  5th,  and  the 
aug.  triad  containing  an  aug.  5th  (hence  their  names). 


68  MUSICOLOGY 

4.  The  dim.  triad  is  a  diatonic  triad  on  7  in  major  keys, 
and  on  7  and  2  in  minor  keys  (see  Fig.  10,  p.  43).  The  aug. 
triad  is  diatonic  only  on  3  of  minor  keys.  These  triads  are 
used  in  all  their  positions  and  inversions — the  dim.  5th  in- 
verted becomes  an  aug.  4th,  and  the  aug.  5th  inverted  be- 
comes a  dim.  4th,  and  therefore  still  dissonant. 

5.  All  chords  not  in  their  diatonic  position  in  a  key  are 
chromatic  chords, as  they  involve  accidental  or  chromatic  tones. 

6.  Extensions  of  the  Triad.  The  triad  is  the  foundation 
of  all  chords,  but  on  the  principle  that  harmony  builds  in  3ds, 
we  may  extend  the  triad  by  adding  3ds ;  but  any  extension 
of  the  triad  will  involve  dissonant  intervals ;  thus,  by  adding 
a  3d  to  the  triad  we  get  the  interval  of  a  7th  from  the  root, 
adding  another  3d  we  have  the  9th  together  with   the  7th,  etc. 

7.  A  chord  containing  the  interval  of  a  7th  is  called  a  chord 
of  the  7th  ;  a  chord  containing  the  interval  of  a  9th  is  called 
a  chord  of  the  9th ;  etc.  Thus  continuing  to  add  3ds,  we 
would  get  the  chords  of  the  i  ith,  13th,  etc. 

8.  A  full  chord  of  the  7th  contains  four  distinct  tones;  a 
full  chord  of  the  9th  contains  five  distinct  tones,  etc.  If  in 
the  chord  of  the  7th  either  the  ist  or  7th  be  omitted,  the  chord 
loses  its  identity  by  becoming  a  triad  ;  or  if  in  the  chord  of  the 
9th  cither  the  ist  or  9th  be  omitted,  the  chord  loses  its  iden- 
tity by  becoming  a  chord  of  the  7th.  The  5th  or  middle  tone 
of  a  chord  of  the  9th  is  usually  omitted  in  four-part  harmony. 

9.  Some  theorists  explain  these  chords  as  combined  triads 
(two  triads  combined  through  their   common  tones).      Thus 

the  triads  ?_,?  combined  through  their  common   tones  would 

4  ^  6  « 

give  2  —  a  chord  of  the  7th  ;    or  g—o  combined    would    give  | 

5  5  6 

—  a  chord  of  the  9th.  A  chord  of  the  iith  would  be  two 
complete  triads,  one  above  the  other.  But  the  simplest 
method  of  treatment  is  that  of  extended  triads. 

10.  As  the  triads  arc  represented  by  the  roman  numerals, 
the  extended   triads  may  be  represented  by  simply  attaching 


STRUCTURE    OF    MUSIC 


69 


a  figure  corresponding  to  the  distinguishing  interval;  thus,  ll^, 
Vy,  or  Vg,  etc. 

II.  Chords  of  the  7th.  A  7th  chord  contains  six  inter- 
vals, which  in  the  direct  close  form  of  the  chord  are :  three 
3ds  one  above  the  other,  two  5ths  interlocked,  and  a  7th 
spanning  all,  thus:  (J)  .  But  a  closer  analysis  (see  major 
and  minor  patterns,  p.  43)  will  show  that  there  are  seven 
kinds  of  7th  chords,  outlined  as  follows: 


Class 

7th  Chords  in  Direct  Close  Form 

Analysis 

1st 

Major  triad  with  major  7th — It  and  IVt 
in  major  keys,  and  VI7  in  minor  keys. 

major  3d  +  minor  3d  +  major 
3d,   two  perfect    stlis    inter- 
locked, major  7th  spanning 
all. 

2d 

Minor  triad  with  minor  7th — Ht,  HI?  and 
VI7  in  major  keys, and  iv,  in  minor  keys. 

minor  3d  +  major  3d  +  minor 
3d,    two  perfect   5ths    inter- 
locked,  minor  7th  spanning 
all. 

3d 

Major  triad  with  minor  7th — ■V^  in  both 
major  and  minor  keys. 

major  3d-4-minor  3d  +  minor 
3d,  perfect  5th  and  dim.  5th 
interlocked,  minor  7th  span- 
ning all. 

4th 

Minor  triad  with  major  7th — It  in  minor 
keys. 

minor  3d  +  major  3d-l-major 
3d,  perfect  5th  and  aug.  5th 
interlocked,  major  7th  span- 
ning all. 

5th 

Dim.     triad     with    minor     7th — vn'-   in 
major  keys,  and  ii"?  in  minor  keys. 

minor  3d  +  minor  3d  +  major 
3d,  dim.  5th  and  perfect    5th 
interlocked,  minor  7th  span- 
ning all. 

6th 

Aug.   triad    with    major    7th — III't    in 
minor  keys. 

major  3d  +  rTiajor  3d+minor 
3d,  aug.   5th  and  perfect  5th 
interlocked,  major  7th  span- 
ning all. 

7th 

Dim.  triad  with  dim. 7th — viT-   in  minor 
keys. 

Three  minor  3ds,  one  above 
the  other,  two  dim.  5ths  in- 
terlocked, dim.  7th  spanning 
all. 

Class— 
7th  Chords  in  Major  Keys 

ist 

I; 

2tl 

11; 

2d 

ist 
IVt 

3d 
Vt 

2d 

VI7 

5th 
VI T- 

Class- 
7th  Chords  in  Minor  Keys 

4tl. 

It 

stli 

t,th 
III't 

2d 

IVt 

Vt 

ist 

VIt 

7th 

VII  T 

70  MUSICOLOGY 

12.  Observe,  as  to  the  number  of  dissonant  intervals  con- 
tained in  each  class :  the  1st  contains  one  (major  7th);  the 
2d,  one  (minor  7th);  the  jd,  two  (dim.  5th  and  minor 
7th);  the  7///,  two  (aug,  5th  and  major  7th);  the  jM,  two 
(dim.  5th  and  minor  7th);  the  6tJi,  two  (aug.  5th  and  major 
7th);  the  ytJi,  three  (two  dim.  5ths  and  a  dim.  7th) — thus 
giving  a  comparative  idea  of  their  dissonant  character,  although 
some  dissonant  intervals  are  more  dissonant  than  others. 

13.  To  become  more  familiar  w^th  the  structure  of  these 
chords  by  comparisons,  we  may  make  the  following  observa- 
tions : 

First,  that  the  jc/  and  5///,  also  the  ^tJi  and  6t]i  classes 
are  composed  of  the  same  intervals,  but  in  reverse  order. 

Second,  that  in  the  ist  and  2d  classes  the  intervals  that 
are  major  in  one  are  minor  in  the  other. 

Third,  that  major  3d  +  minor  3d  =  perfect  5th  ;  minor 
3d  +  minor  3d  =  dim.  5th  ;  and  major  3d  +  major  i([  =  ''^>^'g- 
5th. 

Fourth,  that  IV,  and  VI.,  vil°  and  II7,  II .  and  IV,,  are, 
each  pair,  one  and  the  same  chord  in  relative  major  and  minor 
keys  (Fig.  10,  p.  43),  and  that  V,  is  one  and  the  same  chord 
in  tonie  major  and  minor  keys. 

h'ifth,  that  the  stroke  under  Vll°  in  minor  keys  is  to  distin- 
guish it  from  Vll°    in  major  keys. 

Sixth,  that  the  ist  and  2d  classes  occur  more  than  once 
in  any  major  key,  and  the  other  classes  occur  only  once  in 
any  key. 

Seventh,  that  the  form  of  the  vii,  is  not  changed  by 
inverting  it;  as  the  3ds  arc  equal,  and  each  equal  to  the 
augmented  2d  (i^  steps),  and  together  equal  an  octave,  and 
therefore  form  the  same  intervals  in  any  order  (see  minor 
pattern). 

14.  The  inverted  forms  of  these  chords  (except  the  VII^) 
would,  of  course,  involve  different  combinations;  as  3ds 
invert  into  6ths,  5ths  into  4ths,  and  7ths  into  2ds  (major  into 


STRUCTURF,    OF    MLTSIC 


71 


minor  and  vice  versa,  dim.  into  aug.  and  vice  versa,  and  per- 
fect into  perfect),  but  dissonant  intervals  invert  into  disso- 
nant intervals,  and  consonant  into  consonant,  so  that  the  dis- 
sonant character  of  the  chords  is  but  little  changed. 

15.    The  intervals  do  not  all  invert  at  once,  but  in  the  order 
shown  in  Fig,  15. 


Fig.  ir,. 


16.  The  chords  of  the  7th,  with  their  inversions,  are  all 
more  or  less  used,  but  some  are  much  more  prominent  and 
important  than  others.  Those  of  special  importance  are  the 
V,  (called  the  Dominant  7th)  and  the  vil°  (called  the  Dimin- 
ished 7th)  chords.  These  are  the  principal  chords  used  in 
modulation,  and  will  be  more  fully  described  under  ''Modu- 
lation." 

17.  Positions  and  Forms  of  the  7th  Chords.  A  chord  of 
the  7th  has  four  positions,  as  each  tone  may  be  in  the  highe.st 
voice;  also,  four  forms  (direct,  and  three  inversions),  as  each 
tone  may  be  in  the  bass. 


2d          3d         4th 
pos.      pos.      pos. 

1       1      -_  __ 

I  St 

pos. 

3d        4th 
pos.       pos. 

ist        2d 
pos.      pos. 

1 

4th 

pos. 

ist 
pos. 

2d 

pos. 

1 

3(1 
pos. 

1 

rP"        '        *'        i 

i         d         m    ' 

1 

J 

J 

1         *         ' 

J 

I            .J            S            1 

CS— 

— 3 a r — 

—'r 

— ^# — » — 

-^# — U — i~ 

— «— 

— a— 

-i i 

Direct  form 

^WT- 

ist  Inv. 

-0- 

2d  Inv. 
1             1 

__l 

t 

• 

3d  Inv 

fm\'        P           P          P 

1                      )                      1 

1             ' 

•        '        '        ' 

\fij'        ill 

1           ' 

J           J 

^ 

^ 

r         1 

CS' 

J 

^           ^ 

1                  1                  1                  1 

1              1              1              1 

5  5  S3  3  3 

Showing  the  positions  and  forms  of  the  V7  chord  in  the  key  of  C. 
Fir..  10. 


72  MUSICOLOGV 

1 8.  Thorough-bass  Figuring  of  7th  Chords.  Taking  the 
V,  chord  as  an  example  and  figuring   the   inter\-als    from   the 

bass,  we  find  that  when  the  bass  is  the  root  I  f  I  the  intervals 
of  the  other  parts  from  the  bass  are  ^'h  —  the  figure  7  repre- 
sents the  chord  in  direct  form.  In  the  ist  inversion  (i)  the 
intervals  would  be  |;|'  —  5  shows  1st  inversion.  In  the  2d 
inversion  1 1  \  the  intervals  would  be  fj  —  f  shows  2d  inversion. 
In  the  3d  inversion  (  1 1  the  intervals  would  be4t'h  —  ^,  or  simply 

2,  shows  3d  inversion. 

19.  It  will  be  noticed  that  the  abbreviated  figuring  of  each 
form  represents  the  intervals  peculiar  to  that  form  ;  the  other 
intervals,  which  are  always  built  in  3ds  and  therefore  implied, 
are  omitted  for  the  sake  of  brevity.  All  7th  chords  are  fig- 
ured the  same  way  ;  therefore,  these  figures  are  only  to  show 
the  form  of  the  chord  (whether  direct,  ist,  2d,  or  3d  inver- 
sion).     (See  I^^ig.   16.) 

20.  The  Chord  of  the  9th  (figured  ?).  The  chords  of  the 
9tli,  iith,  etc.,  are  used  practically  only  on  the  dominant. 
(When  on  any  other  note  than  the  dominant,  Qths,  i  iths, 
etc.,  are  usually  regarded  as  suspensions.) 

21.  The  full  chord  of  the  9th  in  its  direct  close  form  con- 
sists of  ten  intervals,  anah'zed  in  major  keys,  thus: 

/f"v\.  j  Major  3ci  +  minor  3d  +  minor  3d  +  major  3d. 

(a.  f\\  I  Two  perfect  5ths  linked  together  by  dim.  5th. 

Vg       fvi?//  I  Two  minor  7ths  interlocked. 

Vr//  1  ■■^  major  9th  spanning  all. 

It  cont.iins,  therefore,  four  dissonant  intervals  (dim.  5th, 
two  minor  /ths,  and  a  major  9th).  In  minor  keys  it  would 
contain  five  dissonant  intervals  (two  dim.  5ths,  minor  7th, 
dim.  7th,  and  a  minor  9th).  It  is  used  in  its  ist,  2d,  and  3d 
inversions. 

22.  If  the  root  of  the  Y ,j  chord  is  omitted,  it  becomes 
either  the  Vii!   or  vii°.   chord,  according  as  the  9th   is   major 


STRUCTURE    OF    MUSIC  73 

or  minor.  As  these  two  chords  have  no  perfect  5th,  and  on 
the  principle  that  every  fundamental  chord  should  have  a 
perfect  5th,  some  theorists  regard  them  as  V^  chords  with 
root  omitted,  thus  including  them  in  the  general  class  of 
dominant  harmony. 

23.  Chords  of  the  6th.  From  Fig.  15  (p.  71)  we  see  that 
a  chord  of  the  7th  (like  the  triad)  inverts  into  a  chord  of  the 
6th  (spanned  by  a  6th),  and  the  first  inversion  has  the  appear- 
ance of  being  a  triad  with  a  6th  added,  so  that  Fig.  15  may 
be  regarded  either  as  a  chord  of  the  7th  with  its  inversions  or 
as  a  chord  of  the  6th  with  its  inversions  (the  direct  form  of 
the  7th  chord  being  the  3d  inversion  of  the  6th  chord);  but 
on  the  principle  that  harmony  builds  in  3ds,  it  is  more  cor- 
rect to  regard  this  as  a  chord  of  the  7th.  However,  the  1st 
inversion  of  the  7th  chord,  when  it  consists  of  a  major  triad 
with  a  major  6th  (as  in  the  ist  inv.  of  the  2d  class,  p.  69), 
it  is  often  called  a  chord  of  the  Added  6th. 

24.  In  the  analysis  of  the  2d  class  (p.  69),  we  see  that  the 
three  upper  parts  of  the  7th  chord  form  a  major  triad  as 
against  the  minor  triad  formed  by  the  three  lower  parts  (re- 
garding the  7th  chord  as  two  overlapping  triads) ;  and  in  view 
of  the  stronger  and  more  aggressive  character  of  the  major 
triad,  the  ist  inversion,  which  places  it  below  as  the  founda- 
tion, may  therefore  be  regarded  as  the  fundamental  position 
of  the  chord  ;  hence,  the  justification  of  the  added  6th  chord. 
Observe  that  this  is  the  case  only  in  7th  chords  of  the  2d 
class  (formed  by  adding  a  minor  7th  to  a  minor  triad). 

25.  The  1st  inversion  of  a  major  triad  is  sometimes  called 
the  Neapolitan  6tJi.  It  consists  of  a  tone  with  its  minor  3d 
and  minor  6th. 

26.  The  Augmented  6th  chord  is  a  chord  containing  an  aug. 
6th.  By  examining  the  major  and  minor  patterns,  it  will  be 
seen  that  the  aug.  6th  is  not  a  diatonic  interval  of  either.  It 
is  formed  only  by  augmenting  a  major  6th,  and  is  therefore 
a  chromatic   interval,  and  the  aug.  6tli  chord    is  therefore   a 


74  MUSICOLOGY 

chromatic  chord  in  any  key.  It  exists  in  three  forms:  Ital- 
ian 6th,  French  6th,  and  German  6th.  (These  are  fully  de- 
scribed on  p.  104.)  All  other  chords  are  diatonic  in  some  key, 
and  when  used  in  other  keys  as  chromatic  chords,  they  have 
a  tendency  to  lead  to  the  key  in  which  they  are  diatonic,  or 
natural,  thus  tending  to  induce  modulation  (change  of 
key). 

INTERRELATIONSHIP  OF  CHORDS  IN  GENERAL 

1.  Chords  may  be  related  in  three  ways:  first,  through  be- 
longing to  the  same  diatonic  scale  (common  key  relationship) ; 
second,  through  the  relationship  of  their  roots  (root  relation- 
ship); third,  through  common  tones  (common  tone  relation- 
ship). 

2.  Common  Key  Relationship.  A  key  is  a  family  of  tones 
related  through  belonging  to  the  same  diatonic  scale ;  there- 
fore, chords  belonging  to  the  same  diatonic  scale,  or  key,  are 
related  as  being  composed  of  interrelated  tones.  Naturally, 
also,  those  chords  based  on  the  most  prominent  divisional 
points  of  the  scale  (1,4,  5)  are  most  prominently  related  to 
each  other  through  their  common  relationship  to  the 
key. 

3.  Chords  thus  related  through  belonging  to  the  same  dia- 
tonic scale  are  naturally  associated  together;  hence  their  ten- 
dency (when  used  in  other  keys  as  chromatic  chords)  to  lead 
to  the  key  in  which  they  are  diatonic. 

4.  But  certain  keys  (especially  those  closely  related)  have 
a  number  of  tones  in  common,  and  have  therefore  certain 
chords  in  common,  but  occupying  different  positions  (with 
different  roman  numeral  name)  in  the  different  keys.  These 
common  chords  naturally  form  connecting  links  between  the 
keys  containing  them,  and  are  convenient  stepping-stones  in 
passing  out  of  one  key  into  the  other.  They  arc  called  equiv- 
ocal chords,  because,  belonging  to  different  diatonic  scales, 
they  lead  to  no  particular  key,  but  to  any  one  of  the  keys  in 


STRUCTURE    OF    MUSIC  75 

which  they  are  common,  while  those  chords  which   belong  to 
only  one  diatonic  scale  always  lead  to  that  particular  key. 

5.  Root  Relationship.  In  this  sense,  chords  are  related  in 
proportion  to  the  consonance  of  the  interval  between  their 
roots.  If  the  interval  is  a  perfect  concord  (octave,  perfect 
5th,  or  perfect  4th),  we  have  the  1st  degree  of  relationship. 
If  the  interval  is  an  imperfect  concord  (major  or  minor  3d  ox 
6th),  we  have  the  2d  degree.  If  the  interval  is  a  dissonance, 
we  have  the  3d  degree.  When  the  interval  is  an  octave,  the 
relationship  is  so  close  that  the  chords  are  regarded  as  one 
and  the  same. 

6.  Common  Tone  Relationship.  Keys  are  naturally  related 
in  proportion  to  the  number  of  tones  in  common  ;  and,  on 
the  same  principle,  chords,  whether  belonging  to  the  same 
key  or  different  keys,  are  related  in  proportion  to  the  number 
of  tones  in  common. 

7.  In  view  of  the  different  kinds  of  relationship,  the  grade 
of  relationship  between  chords  depends  upon  the  number  of 
points  of  relationship  they  contain. 

INTERRELATIONSHIP   OF    THE    TRIADS   OF   A   KEY 

1.  As  the  triads  are  the  foundations  of  all  chords,  it  is  well 
to  notice  their  relation  to  each  other  in  the  same  key. 

2.  The    triads  1 1,  III  I,  V?,  vii°t  ill,  IV  S,  vi  ?,  U,  have 

1  3         5'  7         a  ,4  6        1 

each  two  tones  in  common  with  those  on  either  side  (form- 
ing a  complete  circle  of  the  triads),  and  may  be  called  double- 
connected  (abbreviated  dc)  triads.  Thus  we  see  that  triads 
whose  roots  form  the  interval  of  a  3d  are  iff  triads. 

3.  The    triads  IL  V?,  ut,  vi  ?,  ml,  vn°i  IV  J,  I  g,  have 

^  15263  741 

each  one  tone  in  common  with  those  on  either  side  (also 
forming  a  complete  circle  of  the  triads),  and  may  be  called 
single-connected  (abbreviated  sc)  triads.  Thus  we  see  that 
triads  whose  roots  form  the  interval  of  a  5th  are  sc  triads. 

4.  Triads  whose  roots  form  the  interval  of  a  2d  have  no 
tones  in  common. 


76  MUSICOLOGY 

5.  The  common  tone  relationship  of  the  triads  may  be 
illustrated  as  in  Fig.   17,  showing  the  circle  of  triads  with  the 

dc  triads  connected  by  double  lines, 
and  the  sc  triads  by  single  lines. 
These  common  tones  are  called  con- 
necting or  binding  tones. 

6.    We  see  that  dc  triads  are  more 
/iqi   closely  related  from  the  view  of  com- 
mon tone  relationship,  and  the  sc  tri- 
ads are  more  closely  related  from  the 
^  ^S  view  of  root  relationship  (except  be- 

FiG.  ir.  tween  diminished  or  augmented  triads 

and  the  triad  on  the  5th  above,  where  the  interval  between  the 
roots  is  a  diminished  or  an  augmented  5th);  and  therefore 
the  actual  relationship  in  the  two  cases  is  about  equal.  The 
very  close  interrelationship  of  the  triads  I,  IV,  V,  is  mainly 
due  to  their  occupying  the  places  of  special  prominence  in 
the  same  key. 

7.  The  triads  of  minor  keys  are  also  related  in  a  precisely 
similar  way. 

PROGRESSION 

1.  Progression  is  passing  from  one  chord  to  another.  Mere 
change  of  position  or  inversion  of  the  same  chord  is  not  pro- 
gression. A  chord  may  pass  through  several  positions  or  in- 
versions before  progressing  to  the  next  chord. 

2.  The  interrelationship  of  chords  forms  the  natural  means 
of  progression,  as  chords  naturally  lead  to  related  chords; 
therefore  the  closer  the  relationship  between  chords  the 
more  natural  the  progression  from  one  to  the  other,  but  va- 
ried relationship  is  necessary  to  varied  expression. 

3.  If  we  analyze  a  sentence  we  find  that  all  the  words  are 
grammatically  related ;  similarly  in  a  musical  sentence,  the 
chords  should  be  musically  related.  For  special  effect,  chords 
are  sometimes  used  without  musical  connection,  just  as,  for 
the  same  purpose,  words  are  sometimes  used  without  gram' 


STRUCTURE    OF    MUSIC  jy 

matical  connection  ;    but  in  cither  case,  this  is  the   exception 
and  not  the  rule. 

4.  The  common  or  connecting  tones  are  the  natural  step- 
ping-stones in  passing  out  of  one  chord  into  another,  and  nat- 
urally, the  more  tones  in  common  the  less  motion  involved 
and  therefore  the  smoother  the  progression.  On  the  other 
hand,  the  more  motion  involved  the  livelier  the  progression. 

5 .  When  a  chord  progresses  according  to  its  natural  tendency, 
it  is  said  to  resolve.  The  words  progression  and  resolution 
are  therefore,  to  a  certain  extent,  synonymous;  but  in  so  far 
as  we  may  draw  a  dJ\'s,\\x\.Q.\\ov\.^ progression  applies  to  concords, 
or  tones  free  to  move  in  any  direction,  having  no  special  ten- 
dency ;  and  resolution^  to  discords  and  leading  tones,  which 
have  a  tendency  to  move  in  a  certain  direction.  Therefore, 
progression  deals  more  especially  with  the  triad,  or  consonant 
part  of  chords ;  and  resolution,  with  the  dissonant  part  of 
chords  and  leading  tones. 

6.  Progression  of  Triads.  Each  tone  of  a  triad  or  other 
chord  is  regarded  as  a  voice ;  and  Voiee  Leading  consists  in 
leading  each  voice  of  one  chord  to  its  proper  place  in  the  next 
chord. 

7.  General  Rule  for  Progression,  or  Voice  Leading : 
(i)  Keep  the  connecting  tones,  if  any,  each  in  the  same  voice  in 
both  triads;  (2)  Lead  the  other  voiee  or  7'oices  one  degree  up  or 
down  {as  the  case  may  be),  each  to  the  nearest  tone  i)i  the  sec- 
ond triad. 

8.  It  will  be  seen  that  the  rule  involves  the  least  motion 
possible.  If  the  rule  is  strictly  followed,  we  have  strict  voice 
leading;    otherwise,  free  voice  leading. 

9.  It  may  be  shown  that  from  any  triad  we  can  reach  any 
other  triad  without  moving  any  tone  more  than  one  degree. 
Thus,  taking  any  triad  for  example,  as  1 1  15^  ist  position  I. 
Now  if  all  three  of  the  tones  ascend  one  degree,  we  have  the 
triad   II  I  zbt  ist  pos.  j;    if  all   three   descend    one   degree,  we 

have  triad  Vll"^  (  riz   ist  pos.  I;    if  the  tones  %  ascend   one  de- 


78  MUSICOLOGY 

gree,  we  have  the  triad  IV  (ij-  3tl  pos.  1;  if  the  tones  f  de- 
scend one  degree,  we  have  the  triad  V  (  ^5^  2d  pos.  );  if  the  tone 
5  ascends  one  degree,  we  have  the  triad  V^I  (  zhz  2d  pos,  j;  if 
the  tone  i  descends  one  degree,  we  have  the  triad  lll/i]z  3d 

pos.  \.  Thus  we  hax-e  passed  from  I  direct  to  each  triad  of  the 
key. 

10.  We  notice  also  that  the  different  positions  of  chords 
are  the  necessary  result  of  voice  leading  where  some  of  the 
parts  arc  held  as  connecting  tones. 

11.  Progression  by  sc  and  dc  triads  difTers  only  in  abrupt- 
ness. Thus  the  progression  I  to  V  may  be  made  direct,  as 
ifigi,  moving  both  tones  at  the  same  time  ;  or  through  the 
intermediate  (^c  triad  III,  as  ifi£r2z,  moving  the  same  tones 
one  at  a  time.  The  progression  IV  to  V  may  be  made  direct 
(moving  all  three  tones  at  the  same  time),  or  through  the  sc 
triad  I,  or  through  the  ^/c  triads  II  and  Vll°. 

12.  The  characteristic  triads  I,  IV,  V,  on  account  of  their 
prominence,  form  the  greater  part  of  any  piece  of  music,  and 
the  progression  from  one  to  the  other  is  frequently  direct ; 
but  we  see  that  dc  triads  may  be  used  between  to  make  the 
progression  smoother. 

13.  It  should  be  remembered  that  the  rule  for  progression 
applies  only  in  passing  from  one  triad  to  another  of  different 
name,  but  does  not  apply  in  passing  from  one  position  to 
another  of  the  same  triad  (which  is  not  progression). 

14.  Rule  for  Skips  :  Consonant  intervals  may  progress  by 
skips,  but  (iisso)iant  interi'als  should  be  approaeJied  and  quitted 
witJioiit  skips.  The  naturalness  of  a  skip  is  in  proportion  to 
the  consonance  of  the  interval  taken. 

15.  The  different  positions  of  the  same  triad  ma}-  be  used 
freely,  as  only  consonant  intervals  are  involved  ;    but   in  pro- 


STRUCTURE    t)K    MUSIC 


79 


crressins  to  a  new  triad  with  one  or  more  tones  dissonant  to 
the  first  triad,  it  is  best  to  do  so  with  the  least  motion  possi- 
ble, by  holding  the  connecting  tones  and  moving  the  others 
without  skips  to  the  nearest  tones  of  the  new  triad.  There- 
fore skips  most  usually  occur  in  repeating  same  triad  in  a  dif- 
ferent position.  When  chords  progress  by  skips,  without 
any  definite  connection,  they  are  called  leaping  or  jumping 
chords. 


ffi 


^s 


£: 


T=^ 


:e=J: 


Fig.  18. 

16.  Fig.  18  illustrates  the  rule  of  progression.  It  will  be 
observed  that  the  rule  applies  only  to  the  three  upper  parts 
(triad  proper),  but  does  not  apply  to  the  bass. 

17.  The  Progression  of  the  Bass  involves  the  same  prin- 
ciple (relationship  of  intervals)  as  the  root  relationship  of 
chords.  The  natural  tendency  of  the  bass  is  to  progress  by 
consonant  skips ;  the  more  perfect  the  interval  of  the  skip, 
the  more  natural  the  progression  :  therefore  the  most  natural 
progressions  of  the  bass  are  the  perfect  concords  (octave,  per- 
fect 5th,  and  perfect  4th),  after  these  the  imperfect  concords 
(major  and  minor  3ds  and  6ths),  and  lastly,  the  single  dia- 
tonic degree  or  2d. 

18.  The  bass  is  usually  a  doubling  of  one  of  the  tones  of 
the  triad  above  it.  Its  progression  is  therefore  influenced  by 
both  considerations.  Besides,  if  the  bass  progressed  like  the 
three  upper  parts  according  to  the  rule,  we  would  find  it  im- 
possible to  avoid  consecutive  octaves  and  perfect  5ths,  while 
the  tendency  of  the  rule,  as  between  three  parts,  is  to  avoid 
them. 


8o  IMUSICOLOGY 

19.  Rule   Regarding   Consecutives :    Tzuo  perfect  ^t/is  or 
octaves  should  not  occur  co)iseciitively  between  the  same  parts. 

Consecutive  Fifths  Consecutive  Octaves 


Fig.  19. 


20.   They  are  not  considered  consecutive  unless  they  occur 
between  the  same  parts.      Neither  are  repeated   perfect   5ths 

and  octaves  (see  Fig.  20)    con- 

f—~\--    |— -|     sidered  consecutives,  as  they  are 

merely    repetitions  of  the  same 


1i=t 


p,(-  20.  tones  and  equivalent  to  holding 

two  tones. 

2 1 .  The  general  reason  for  the  objection  to  consecutive 
perfect  5ths  or  octaves  is  due  to  the  sense  of  perfectness 
which  each  carries  in  itself,  and  therefore  want  of  connection 
with  each  other,  so  that  the  sense  of  relationship  between  the 
consecutive  chords  is  lacking. 

22.  The  special  reason  for  the  objection  to  consecutive  per- 
fect 5ths  is  due  to  the  tendency  of  the  perfect  5th,  when  used 
consecutively,  to  carry  with  it  its  key  limitations,  thus  chang- 
ing the  key  with  each  progression,  often  giving  to  the  music 
the  effect  of  being  in  more  than  one  key  at  the  same  time. 

23.  The  special  reason  for  the  objection  to  consecutive 
octaves  (also  unisons)  is  that  they  destroy  the  indi\iduality 
of  the  parts  moving  in  octaves  (or  unison),  since  two  parts 
moving  together  in  octaves  (or  unison)  blend  so  as  to  sound 
like  one  part ;  and  when  the  individuality  is  thus  destroyed 
for  a  few  chords  only,  it  gives  the  impression  of  an  accidental 
bad  progression.  This  is  not  the  case  when  the  parts  do  not 
pretend  to  vary,  as  when  two  or  more  voices  sing  the  same 
part  in  unison  or  octaves ;  thus  when  a  man  and  woman  sing 
the  same  part  together,  they  naturally  sing  in  octaves  with 
evidently  no  disagreeable  effect.      They  could  not,  however. 


STRUCTURE    OF    MUSIC  8 1 

sing  together  in  perfect  5ths  without  giving  the  effect  of  each 
being  in  a  different  key. 

24.  Consecutive  perfect  5ths  and  octaves  are  sometimes 
hidden  (heard  but  not  seen).  Hidden  consecutives  occur 
when  two  parts  move  by  similar  motion  to  a  perfect  5th  or 
octave,  as  we  mentally  pass  through  the  intermediate  tones 
which  involve  the  perfect  5th  or  octave. 

25.  Consecutive  faults  can  occur  only  in  similar  or  parallel 
motion,  hence  oblique  motion  should  be  used  as  much  as  pos- 
sible to  avoid  them.  They  are  most  noticeable  betw^een  the 
outside  (most  prominent)  parts ;  between  the  inside,  and  be- 
tween inside  and  outside  parts,  they  are  sometimes  allowed. 

26.  Rule  Regarding  False  Relations  :  Fa/se  relations  should 
be  avoided.  A  false  relation  is  where  a  false  (imperfect) 
prime,  octave,  or  5th  is  formed  between  two  difTerent  voices 
in  consecutive  chords;  thus,  if  a  voice  in  one  chord  sings  C 
and  another  voice  in  the  next  chord  sings  C  i|,  a  false  prime 
or  octave  relation  is  formed  ;  or  if  a  voice  in  one  chord  sings 
B  and  another  voice  in  the  next  chord  sings  F,  a  false  5th 
relation  is  formed — but  the  same  voice  may  sing  both  notes. 
False  5th  relations  are  forbidden  only  between  the  outside  parts. 

27.  A  false  prime  or  octave  relation  involves  two  keys, 
since  the  two  voices  are  not  in  the  same  key;  but  if  the  same 
voice  sings  both  notes,  the  second  note  is  merely  of  the  na- 
ture of  a  passing  note.  The  same  in  effect  is  also  true  of  the 
false  5th  relation. 

28.  Motion  of  the  Parts.  The  progression  of  chords  with 
their  changes  of  position  produce  what  is  called  motion  in  the 
parts.  There  are  three  kinds  of  motion  as  between  any  two 
parts :  Contrary  motion  (when  one  part  ascends  and  the  other 
descends) ;  Oblique  motion  (when  one  part  moves  horizontally 
and  the  other  ascends  or  descends) ;  Similar  motion  (when 
both  parts  ascend  or  descend  together).  Similar  motion  is 
also  called  parallel  motion  when  the  parts  move  parallel  to 
each  other. 


82  MUSICOLOGV 

29.  Preparation  of  Dissonances.  A  dissonant  tone  is  said 
to  be  prepared  when  it  appears  (in  the  same  voice)  as  a  con- 
sonant tone  in  the  preceding  chord. 

30.  As  a  rule,  every  dissonant  tone,  except  the  do)niiiant  jtJi 
and  diiiiinisJied  jtJi  {zvJiich  juay  or  may  not  be  prepared)  and 
passing  notes,  should  be  prepared  by  being-  first  heard  as  a  eon- 
sonant  tone  before  it  is  heard  as  a  dissonajit  tone.  Dominant 
/ths  and  diminished  7ths  have  become  so  familiar  as  to  cease 
to  require  preparation.  It  is  difficult  for  the  voice  properly 
to  intone  a  dissonance,  but  when  first  intoned  as  a  consonance 
it  is  easy  to  hold  ;   hence  the  rule. 

RESOLUTION 

1.  Resolution  in  its  general  sense  applies  to  any  tone  or 
chord  that  progresses  according  to  its  natural  tendency. 
Resolution  therefore  is  merely  natural  progression.  Any 
chord  (if  it  cannot  remain  stationary)  tends  to  move  to  the 
chord  that  occasions  the  least  motion,  using  the  common 
tones  as  connecting  links.  The  rule  for  progression  (p.  /J:/) 
is  based  upon  this  tendency;  therefore  chords  which  pro- 
gress strictly  according  to  the  rule  may  properly  be  said  to 
resolve. 

2.  The  key-note,  or  tonic  (i),  is  the  tone  of  complete  re- 
pose or  home  feeling  in  any  key,  and  is  therefore  the  tone  to 
which  all  other  tones  tend  for  final  resolution.  Next  to  the 
tonic  in  restful  character  are,  first  the  5th,  then  the  3d.  All 
other  tones  are  restless  and   tend  to  resolve   to   one    of   these 

13  5  13  5 

three  restful  tones  (Do,  Mi,  Sol,  in  major  keys;  La,  Do,  Mi, 
in  relative  minor  keys"). 

3.  The  Tonic  triad  being  composed  of  these  three  restful 
tones  is  the  chord  of  final  resolution  of  all  other  chords,  and 
is  the  natural  beginning  as  well  as  ending  of  all  music ;  hence 
all  music  should  begin  and  end  with  the  Tonic  triad,  as  well 
as  frequently  return  to  it  (to  prevent  weariness)  during  the 
music. 


STRUCTURE    OF    MUSIC  83 

4.  As  the  bass  is  the  foundation,  it  should  begin  and  end 
on  the  tonic.  If  the  soprano  also  ends  on  i  (in  octave  or 
octaves  with  the  bass),  it  is  called  a  perfect  close ;  but  if  the 
soprano  ends  on  3  or  5,  it  is  called  an  imperfect  close.  The 
soprano  may  begin  on  i,  3,  or  5. 

5.  As  already  observed,  resolution  in  its  strict  sense  ap- 
plies to  discords  and  leading  tones;  and  progression,  to  con- 
cords (discords  resolve,  concords  progress). 

6.  Dissonant  intervals  have  a  determined  resolution  (ten- 
dency to  move  in  a  certain  direction),  while  consonant  inter- 
vals are  said  to  be  free  (without  tendency  to  move  in  any 
particular  direction).  2ds,  7ths,  Qths,  etc.,  all  diminished 
and  augmented  intervals,  and  perfect  4ths  (when  just  over 
the  bass)*  are  classed  as  dissonant  intervals. 

7.  A  dissonance,  or  discord,  is  caused  by  a  tone  forming  a 
dissonant  interval  with  some  other  tone,  and  the  tone  thus 
causing  the  dissonance  is  called  a  dissonant  tone. 

8.  All  rules  for  Resolution  are  based  on  natural  tendency, 
and  the  natural  tendency  is  to  resolve  in  the  easiest  way.  A 
dissonant  interval  is,  in  a  sense,  an  unnatural  interval,  and 
most  naturally  tends  to  resolve  (or  melt  as  it  were)  into  the 
nearest  consonant  interval  as  being  within  easiest  reach.  For 
the  same  reason,  a  dissonant  interval  is  easiest  reached  from 
the  nearest  consonant  interval.      Hence  the  general  rule: 

Rule  I.  Dissonances  should  be  approached  and  quitted  con- 
junctly {by  the  single  step  of  a  2d ;  i.e.,  without  skip). 

9.  It  is  usually  easier  for  the  voice  to  descend  than  to 
ascend,  as  descending  tends  toward  relaxation,  while  ascend- 
ing tends  toward  tension.  For  which  reason,  it  is  also  a  gen- 
eral rule  : 

*  The  dissonance  of  the  perfect  4th  is  only  relative,  as  the  perfect  4th  in  itself  is 
always  a  consonant  interval.  But  its  natural  relation  in  the  key  is  as  an  inverted  per- 
fect 5th  ;  and  when  used  just  over  the  bass,  it  is,  in  a  sense,  as  a  substitute  for  the  per- 
fect 5th,  and  thus  antagonizes  the  natural  key  relationship,  and  is  therefore  relatively 
dissonant  to  the  key— having  the  effect  of  a  tone  foreign  to  the  key,  and  tending  to  in- 
duce change  of  key.    Therefore,  when  thus  used,  it  must  be  treated  as  a  dissonance. 


^4  MUSICOLOGY 

Rule  II.  Dissonances  tend  to  resolve  doivn,  unless  by  resolv- 
ing up  they  ean  do  so  by  a  smaller  interval  {half-step  instead 
of  a  zvhole  step).  As  a  result  of  this  rule,  7ths,  9ths,  etc., 
diminished  intervals,  and  perfect  4ths  (when  just  over  the 
bass)  tend  to  resolve  down,  while  leading  tones  and  aug- 
mented intervals  tend  to  resolve  up. 

10.  In  view  of  the  tendency  to  maintain  the  key  relation- 
ship, it  is  also  a  rule  : 

Rule  III.  Dissonances  tend  to  resolve  by  steps  and  half- 
steps  of  the  diatonic  scale  of  the  key.  In  a  general  sense,  how- 
ever, dissonances  tend  to  move  by  half-steps  as  the  smallest 
interval  in  use,  and  therefore  by  ignoring  the  key  boundary 
they  may  easily  be  made  to  resolve  chromatically  into  a  new 
diatonic  scale,  or  key,  when  by  so  doing  a  proper  chord  is 
formed  in  the  new  key, 

11.  Rule  IV.  A  dissonant  to7ie  should  not  be  heard  at  the 
same  time  ivith  the  tone  to  luhich  it  resolves.  This  rule  gen- 
erally causes  the  interval  of  a  2d  to  expand  (its  upper  limit 
ascending  or  its  lower  limit  descending).  It  is  also  the  princi- 
pal reason  why  the  7th  descends,  as  otherwise  it  would  resolve 
to  the  octave  of   the  root  (which  is  harmonically  the  same). 

12.  A  chord  to  which  a  discord  resolves  is  called  its  chord 
of  resolution,  and  the  note  to  which  a  dissonant  tone  resolves 
is  called  its  note  of  resolution.  If  the  resolution  of  a  disso- 
nant tone  is  retarded  by  one  or  more  notes  intervening,  it  is 
called  an  ornainental  resolution. 

13.  Dissonant  tones  are  pleasing  if  short,  not  too  frequent, 
and  properly  resolved.  Dissonant  tones,  by  calling  for  reso- 
lution, express  energy  and  progress,  and  therefore  give  vigor 
to  the  music. 

14.  In  the  extensions  of  the  triad  (chords  of  the  7th,  etc.), 
the  added  tone  or  tones  often  form  dissonant  intervals  with 
more  tlian  one  tone  of  the  triad  (see  p.  69) ;  but  it  is  only 
necessary  to  apply  the  rules  of   resolution  to  the  dissonant 


STRUCTURE    OF    MUSIC  85 

tone  which  is  the  cause  of  the  several  dissonant  intervals,  as 
the  different  tones  of  the  triad  are,  in  general,  subject  to  the 
progression  of  the  triad. 

15.  In  the  diminished  and  augmented  triads  (p.  67:  3)  the 
dissonant  5th  will  require  resolution. 

16.  Applying  the  rules  of  resolution  and  progression  to  the 

Dominant  7th  chord  /V,,  ^  ):  7,  being  the  leading  tone  of 
the  key,  ascends  to  i  ;  4  (minor  7th)  descends  to  3  ;  let  5 
remain  stationary  and  2  ascend  or  descend  (5  and  2  being 
free),  and  we  have  the  Tonic  chord.  If  we  let  5  ascend  to 
6,  we  will  have  the  Tonic  chord  of  the  relative  minor  (see 
Fig.  10,  p.  43).  If  in  the  V,  chord  of  the  relative  minor  we 
let  5  ascend  to  6,  we  have  the  IV  chord  of  the  relative  major. 
These  are  the  principal  resolutions  of  the  Dominant  7th  (V,) 
chord.  Thus  a  dissonant  chord  may  have  several  resolutions, 
owing  to  the  free  nature  of  some  of  its  parts,  the  dissonant 
parts  resolving  the  same  in  each  case. 

17.  In  a  similar  manner,  the  other  dissonant  chords  may 
be  resolved  by  applying  the  rules  of  resolution  to  the  disso- 
nant parts. 

18.  In  the  I  chord  (2d  inversion  of  a  triad)  the  interval  of 
a  4th  being  just  over  the  bass  tends  to  partake  of  the  nature 
of  a  dissonance,  and  calls  for  resolution.  The  most  satisfac- 
tory resolution  of  the  |  chord  is  to  the  direct  (§)  chord  on  the 
same  bass. 

19.  It  is  possible,  though  not  so  usual,  instead  of  resolving 
the  dissonant  part,  to  hold  it  and  bring  the  other  parts,  which 
form  dissonant  intervals  with  it,  into  consonant  positions;  or, 
which  amounts  to  the  same  thing  (in  the  case  of  the  extended 
triad),  we  may  either  regard  the  lower  three  tones  of  the 
direct  chord  as  the  triad  proper  and  the  upper  tone  or  tones 
as  dissonant  to  be  resolved  (called  treatment  in  the  upper 
part),  or  we  may  regard  the  upper  three  tones  (especially  if 
forming  a  major  triad)  as  the  triad  proper  and  the  lower  tone 


MUSICOLOUV 


cr  tones  as  dissonant  to  be  resolved  (called  treatment   in   the 
lower  part).      Thus  (taking  the  I,  chord  for  example): 


(Trcatmenl  in  the  upper  part) 


13       5       7  13       5       7 

\/  i     \/  / 

13  6  14       6 


0       V 


(Treatment  in  the  lower  pan) 

13       5       7  13 

\/        I        I  \     \.  . 

2  5       7  2       4  7 

It  will  be  noticed  that  the  last  two  forms  are  symmetri- 
cally the  reverse  of  the  first  two.  In  the  first  two,  the  upper 
note  descends  by  reason  of  Rule  IV;  in  the  last  two,  the 
lower  note  ascends  for  the  same  reason. 

20.  These  forms  as  general  patterns  show  the  most  natural 
resolutions  of  all  /th  chords,  except  the  VII^  and  VII°, ,  in 
which  the  root  of  the  chord  being  the  leading  note  ascends 
and  the  5th  (being  diminished)  and  7th  descend,  giving  for 
their  primary  resolution  the  following  form  (see  major  and 
minor  patterns) : 

7        2       4        6 

\     \/     / 
1       3       5 

21.  As  a  general  rule,  the  resolving  parts  resolve  the  same 
in  all  the  different  positions  and  inversions  of  chords,  as  the 
different  positions  and  inversions  are  merely  the  same  tones 
in  different  orders  (counting  octaves  as  replicates  of  the  same 
tone). 

22.  The  rules  of  progression  and  resolution  are  not  to  be 
regarded  as  absolutely  binding  at  all  times.  They  aim  only 
at  a  smooth,  connected,  natural  flow  of  music  ;  therefore  to 
express  effort  and  excitement  or  any  unusual  effect  they  are 
temporarily  set  aside. 


STRUCTURE    OF    MUSIC  8/ 

SUSPENSION    AND   ANTICIPATION 

1.  Suspension  is  the  holding  of  a  tone  of  one  chord  over 
into  the  next  chord  in  which  it  is  dissonant,  thus  producing 
a  momentary  discord. 

2.  Anticipation  is  taking  a  tone  belonging  to  the  following 
chord  and  holding  it  until  the  other  parts  follow  (reverse  of 
suspension). 

3.  The  object  of  suspension  and  anticipation  is  the  closer 
binding  of  chords  by  thus,  in  a  sense,  linking  them  together. 

4.  Three  things  are  to  be  considered  in  suspension :  its 
preparation,  its  entrance,  and  its  resolution. 

5.  The  general  rule  is:  The  preparation  takes  place  on  an 
unaeeented  pulse  ;  the  entrance,  on  the  following  accented  pulse  ; 
a)id  the  resolution,  on  an  unaccented  pulse — in  other  words,  the 
oit ranee  is  accented ;  the  preparation  and  resolution,  unac- 
cented. The  resolution  may  be  ornamental  (one  or  more 
notes  intervening),  and  the  chord  accompanying  the  suspen- 
sion may  progress  in  the  meantime  to  any  chord  containing 
the  note  of  resolution. 

6.  The  principal  suspensions  are :  9  to  8  (a  suspended  9th 
resolving  into  the  consonance  of  an  octave),  4  to  3  (a  sus- 
pended 4th  resolving  to  a  3d),  7  to  8  (a  suspended  7th  re- 
solving to  an  octave). 

7.  Suspension  and  anticipation  may  also  be  double  or  triple 
(occurring  in  two  or  three  parts  at  the  same  time). 

PASSING    NOTES 

1.  Music  is  made  up  of  Harmony  notes  and  Passing  notes. 
Notes  belonging  to  a  chord  formation  are  called  Harmony 
notes.  Notes  added  to  a  triad  forming  chords  of  the  7th, 
9th,  etc.,  though  dissonant,  are  included  with  harmony  notes. 
Suspended  and  anticipated  notes  are  merely  harmony  notes 
held  over  or  taken  in  advance.  These  are  all  essential  to  the 
harmony  and  are  classed  as  Harmony  notes. 

2.  Notes  used   merely  in  passing  from   one   harmony  note 


88  MUSICOT^OGV 

to  another,  but  which  do  not  belont^  to  the  prevailing  chord, 
are  called  Passing  notes.  Notes  used  merely  for  embellish- 
ment (p.  30:6)  are  also  classed  as  Passing  notes. 

3.  Passing  notes  a  3d,  6th,  or  octave  apart  may  be  used 
simultaneously  in  different  parts.  The  chords  thus  formed 
are  called  passing  chords,  being  too  transient  to  be  harmon- 
ically recognized. 

4.  The  true  Passing  note  (used  in  passing  from  one  har- 
mony note  to  another)  comes  in  and  goes  out  without  skip. 
Two  passing  notes  are  thus  used  in  succession  when  the  inter- 
val between  the  harmony  notes  requires.  These  increase  the 
smoothness  of  the  melodic  flow,  and  for  this  purpose  are 
freely  used  in  music,  especially  in  the  melody  part. 

5.  Other  Passing  notes  come  in  by  a  skip  and  go  out  with- 
out skip.      These  are  more  of  the  nature  of  embellishing  notes. 

6.  Other  Passing  notes  come  in  without  skip  and  go  out 
by  a  skip  of  a  3d  in  the  opposite  direction  (often  going  to 
another  passing  note,  which  latter,  of  course,  belongs  to  the 
preceding  class). 

7.  Other  passing  notes  leave  and  return  to  the  same  har- 
mony note.  In  this  case,  and  also  in  the  case  where  the 
passing  note  comes  in  by  skip  and  goes  out  without  skip,  if 
the  passing  note  is  below  the  harmony  note,  it  tends  to  a 
half-step  below,  thus  often  requiring  an  accidental  ^  or  tt  in 
writing  it  where  the  diatonic  interval  is  not  already  a  half- 
step.  Before  the  3d  of  a  chord  or  the  3d  or  5th  of  a  V, 
chord,  the  passing  note  may  be  a  whole  step  below. 

8.  Passing  notes  are  not  prepared  ;  in  short,  are  not  bound 
by  any  of  the  laws  of  harmony  except  that  being  dissonant 
they  require  to  be  resolved. 

PEDAL   PASSAGE 
I .   A  Pedal  Passage  is  a  passage  in  which  the   bass   sus- 
tains the  tonic  or  dominant  (sometimes  both),  while  the  other 
parts  move  independently  of  it.      The  tone  thus  sustained  is 
called  the  pedal  point,  or   tone.     The  passage   should   begin 


.    STRUCTURE   OF    MUSIC  89 

with  the  harmony  of  the  pedal  tone,  frequently  return  to  it, 
and  finally  end  with  it. 

2.  During  the  pedal  passage,  the  part  next  above  the  pedal 
becomes  the  real  base  as  to  chord  relations. 

3.  There  should  not  be  more  foreign  harmony  in  the  pas- 
sage than  the  pedal  tone  is  capable  of  counterbalancing.  The 
tonic  and  dominant  (having  the  most  sustaining  power)  are 
best  adapted  to  form  the  pedal  point. 

4.  If  the  sustained  tone  is  held  by  one  of  the  upper  parts, 
it  is  called  a  stationary  tone,  and  has  not  the  sustaining 
power  peculiar  to  the  bass. 

THE    CADENCE 

1.  The  Cadence  is  the  close  of  a  musical  strain;  its  object 
is  to  form  a  satisfactory  ending.  It  consists,  in  its  regular 
form,  of  the  Tonic  as  the  final  chord  preceded  either  by  the 
Dominant  (V  or  V^)  or  Sub-dominant  chord;  the  first  is 
called  Authentic;  the  second,  Plagal. 

2.  If  the  cadence  ends  with  the  Dominant  or  Sub-domi- 
nant, instead  of  the  Tonic  chord,  it  is  called  a  Half  Cadence. 

3.  The  Authentic  is  the  strongest  and  most  common  form 
of  cadence.  It  is  also  called  a  Perfect  Cadence  when  the  final 
Tonic  is  in  direct  forni  and  ist  position,  thus  having  the  root 
in  both  bass  and  soprano ;  otherwise,  it  is  called  an  Imperfect 
Cadence. 

4.  A  Deceptive  Cadence  is  where  the  Dominant  resolves  to 
some  other  chord  than  the  Tonic,  thus  deceiving  our  expec- 
tations. 

5.  The  cadence  is  sometimes  ornamented  by  runs,  trills, 
etc.,  in  which  case  it  is  usually  called  a  Cadenaa. 

6.  In  music  the  notes  arc  grouped  into  musical  phrases,  the 
phrases  into  musical  clauses  (called  sections),  the  clauses  into 
musical  sentences  (called  periods).  These  are  arranged  rhyth- 
mically as  in  poetry.  Every  musical  clause  or  sentence  ends 
with  some  kind  of  cadence.  The  cadences  should  be  varied, 
reserving  the  most  complete  cadence  for  the  final  ending. 


90  Mrsicoi.OGY 

7.  In  poetry  or  prose  the  divisions  arc  shown  by  punctua- 
tion marks.  In  music  the  punctuation  marks  are  not  written, 
but  are,  in  a  sense,  understood  ;  thus  a  cadence  calls  for  a 
semicolon,  colon,  or  period  (understood),  according  to  its  de- 
gree of  completeness. 

8.  The  phrases  of  a  musical  clause  are  separated  from  each 
other  by  a  slight  unwritten  pause  called  a  caesura,  which  in 
poetry  is  usually  marked  by  a  comma.  This,  however,  does 
not,  as  a  rule,  involve  a  cadence. 

SEQUENCE 

1.  A  Sequence  is  a  succession  of  similar  movements. 

2.  A  Phrase  Sequence  is  a  musical  phrase  repeated  at  a 
higher  or  lower  pitch. 

3.  A  Chord  Sequoice  \^  3.  swcccssion  of  similar  movements 
of  chords  or  groups  of  chords. 

4.  Symmetry  is  the  controlling  object  in  a  sequence  ;  there- 
fore the  rules  of  resolution  are  suspended  when  necessary  to 
preserve  the  symmetry. 

MODULATION 

1.  Modulation  is  passing  from  one  key  to  another  during 
the  course  of  a  piece  of  music.  If  the  modulation  is  very 
brief,  it  is  called  a  digression. 

2.  An  accidental  sharp  (  ^ )  or  flat  (t?)  or  natural  (3)  usu- 
ally means  a  change  of  key  if  a  diatonic  chord  in  a  new  key 
is  thus  formed.  Sometimes,  however,  the  accidental  is  only 
on  a  passing  tone,  and  is  called  a  chromatic  tone. 

3.  An  accidental  necessarily  introduces  a  new  tone  (foreign 
to  the  signature  key),  and  therefore  tends  to  lead  to  that  key 
which  contains  the  new  tone  while  retaining  as  many  tones  of 
the  old  key  as  possible.  As  keys  are  related  to  each  other 
in  proportion  to  the  number  of  tones  the\'  have  in  common, 
therefore  the  nearer  two  keys  are  related,  the  easier  is  the 
modulation  from  one  to  the  other. 


STRUCTURE    OF    MUSIC  01 

4.  The  Combined  Key  Table  (p.  46)  shows  all  the  keys 
(major  and  minor)  in  the  order  of  their  relationship  to  each 
other;  so  that  (in  major  keys)  each  key  is  related  in  the  first 
degree  to  the  nearest  key  on  either  side  of  it,  and  in  the  sec- 
ond degree  to  the  second  key  from  it  on  either  side,  etc. 
The  connecting  lines  between  the  major  and  minor  key  tables 
show  the  order  of  relationship  between  major  and  minor 
keys. 

5.  Modulation  from  any  major  key  to  the  next  key  on 
either  side  or  to  its  relative  minor  is  called  natural  modula- 
tion, because  easy  and  natural  (the  keys  differing  by  only  one 
tone). 

6.  Modulation  to  a  remote  key  is  either  gradual  (when 
made  by  passing  through  the  intermediate  keys)  or  abrupt 
(when  made  by  stepping  over  the  intermediate  keys).  In  any 
case,  however,  the  general  rule  applies,  that  the  last  sharp 
{ivJicthcr  in  signature  or  as  accidental — major  or  minor  keys') 
is  ahvays  on  the  jtJi  of  the  key  {key  letter,  first  letter  above) ; 
and  the  last  flat  is  alivays  on  the  ^th  of  the  key  in  major  keys 
{key  letter,  fourth  letter  belozv),  or  on  the  6th  of  the  key  in 
minor  keys  {key  letter,  third  letter  above).  This  rule  applies 
also  to  naturals  by  remembering  that  a  natural  has  the  effect 
of  a  flat  in  sharp  keys,  and  the  effect  of  a  sharp  in  flat  keys. 

7.  The  rule  already  given  for  reading  accidentals — giving  a 
sharp  effect  to  the  key  syllable  by  using  i  (e),  or  giving  a  flat 
effect  by  using  e  (a)  or  a  (see  p.  21  :23) — is  the  common  rule, 
and  the  most  convenient ;  this,  however,  is  treating  the  acci- 
dentals as  passing  tones.  It  would  be  more  correct,  there- 
fore, when  a  modulation  occurs,  to  use  the  regular  key  sylla- 
bles readjusted  to  the  new  key  letter,  returning  to  the  signa- 
ture key  as  soon  as  the  modulation  ends  (usually  at  the  end 
of  the  measure,  unless  the  accidental  is  repeated  in  the  next 
measure). 

8.  To  illustrate  the  general  principles  of  modulation,  first 
find  an  accidental  in  a  piece  of  music,    then  take  the  major 


92  MUSICOLOGY 

key  table  and  start  from  the  key  in  the  table  corresponding 
to  the  old  key  in  the  music,  and  at  the  point  on  the  staff  at 
which  the  accidental  occurs  in  the  music,  pass  horizontally 
(to  the  right  in  ^'s,  to  the  left  in  t^'s)  till  you  come  to  the  ji| 
or  b  (treating  a  I;  as  a  t?,  in  sharp  keys ;  and  as  a  ^  in  flat 
keys) ;  however,  do  not  pass  a  5  (to  the  right),  as  it  will  in- 
dicate the  relative  minor  of  that  key ;  also,  do  not  pass  a  6 
(to  the  left),  as  it  will  indicate  the  tonic  minor  of  that  key. 
If  more  than  one  accidental  occurs  in  the  music  at  the  same 
time,  take  the  first  key  that  will  satisfy  them  all ;  in  which 
case,  it  will  be  observed  that  the  last  sharp  or  flat  (which  de- 
termines the  new  key)  will  be  in  the  farthest  key  from  the 
starting  key. 

9.  It  is  evident  that  sometimes  the  accidentals  involved  in 
a  modulation  are  not  all  marked  in  the  music,  as  there  may 
not  always  be  notes  on  all  of  the  affected  lines  or  spaces  to 
mark,  but  are  understood  if  not  marked ;  the  last  sharp  or 
flat  of  any  key  necessarily  involving  all  the  other  sharps  or 
flats  of  that  key. 

10.  Wiien  the  modulation  extends  beyond  the  limits  of 
the  key  table,  we  may  continue  around  the  circle  (Fig.  9,  p. 
42)  by  making  the  enharmonic  change  (p.  42:13);  but  it  is 
preferable,  however,  to  extend  the  table  in  the  direction  of 
the  modulation  when  doing  so  will  occasion  fewer  accidentals 
in  writing  the  music.  Extending  the  table  is  merely  for  the 
purpose  of  avoiding  the  enharmonic  change,  and  thus  avoid- 
ing the  accidentals  which  would  be  involved;  but  extending 
the  table  will  involve  double  sharps  or  flats,  as  already  shown 
(p.  52:  41,  42) 

11.  In  a  harmonic  sense,  an  accidental  does  not  by  itself 
produce  modulation,  but  only  when  with  other  tones  a  dia- 
tonic chord  of  some  other  key  is  formed'  the  chord  thus  be- 
ing a  chromatic  chord  of  the  old  key,  but  a  diatonic  chord  of 
the  new  key ;  the  modulation  being  the  result  of  the  ten- 
dency of  chromatic  chords  to  lead  to  the  key  in  which  they 
are  diatonic,  or  natural. 


STRUCTURE    OF    MUSIC  93 

METHODS   OF    MODULATION 

1.  There  are  many  ways  of  modulating,  but  noticing  a  few 
principal  methods  will  be  sufHcient  to  illustrate  the  general 
principles  involved. 

2.  Modulating  by  Means  of  Connecting  Triads.  Examin- 
ing the  key  table  (p,  46)  and  comparing  the  triads  of  the 
different  keys,  we  would  find  that  all  triads  (except  the  aug- 
mented triad)are  common  to  several  keys  (same  tones  though 
the  triads  have  different  names  in  the  different  keys).  These 
may  be  called  connecting  chords,  since  they  connect  the 
keys. 

3.  It  is  evident  that  if  a  chord  is  common  to  any  two  keys 
it  may  resolve  into  either  key.  Therefore  we  may  modulate 
from  one  key  to  another  by  first  striking  a  chord  common  to 
both,  then  resolving  it  into  a  distinguishing  chord  of  the  new 
key. 


.' 

E^      f 

B^ 

c 

F 

R 

c 

d 

G 

a       D      e 

A 

b 

V 

III 

V 

VI 

I 

II 
III 

I 

IV 

VI 

VI 

IV 

II     I 

IV 

7 

V 

V 

VII° 

I 

III 

IV 
V 

VI 
VIT° 

I 
II 

VI 
IV 

I              IV 
IT° 

VI 

Fig.  21. 


4.  Fig.  21  gives  the  triads  of  the  key  of  C  and  the  corre- 
sponding triads  of  each  in  other  keys  (though  differing  in 
name — roman  numeral — yet  composed  of  the  same  tones). 

5.  By  similar  comparisons,  we  would  find  that  the  triads 
of  any  other  major  key  than  C  would  have  the  same  corre- 
sponding triads  in  other  similarly  related  keys ;  so  that  the 
key  letters  at  the  top  of  Fig.  21  may  be  omitted,  thus  mak- 
ing the  figure  general  in  character,  as  it  is  applicable  to  all 
keys  similarly  related. 


94 


MUSICOLOGV 


6.  We  notice  also  that  the  1st,  4th,  and  5th  formulas  (Fig, 
21)  are  similar  (composed  of  the  same  w^?y'(r^;' triads  in  the  same 
order);  and  that  the  2d,  3d,  and  6th  formulas  are  similar 
(composed  of  the  same  viiiio)-  triads  in  the  same  order).  Now 
omitting  the  repetitions  of  the  same  formulas  we  would  have 
Fig.  22. 


_m^ 


V 

I 

IV 

VI 

III 

YI 

n 

I 

IV 

vir 

11° 

Fig.  'ii. 


7.  Making  the  figure  general  by  omitting  the  key  letters 
(since  these  formulas  apply  equally  to  any  other  keys  simi- 
larly related),  we  will  have  three  general  formulas,  which  may 
be  expressed  thus: 


(Major  triad  formula) 


(Minor  triad  formula) 


(Diminished  triad  formuki) 


If^ — t— ^^         IH— Y^— ^r 

V-^^^         (1)        ^~~^I  (2)^-A-^V 

(The  curved  lines  compare  with  the  curved  lines  in  the  key  table,  p.  46.) 
Fig.  23. 

8.  These  formulas  may  be  copied  (exact  dimension.s)  and 
applied  to  the  key  table  (between  the  major  and  minor 
keys)  so  as  to  show  the  keys  related  to  any  key  through  any 
given  triad.  By  placing  the  given  triad  in  line  (vertically) 
with  the  given  key  (using  the  proper  formula),  the  other  triads 
will  fall  in  line  with  the  related  keys  (the  roman  numerals 
showing  the  relation  of  the  triad  to  the  different  keys).  As 
formulas  1,2,  and  3  are  very  simple,  they  can  easily  be  ap- 
plied mentally  instead  of  applying  a  copy. 

9.  We  may  also  observe  that  as  every  major  key  has  three 
major  triads  (I,  IV, V,  p.  63  :  16)  and  every  minor  key  has  two 
major  triads  (V,  VI),  therefore  every  major  triad    is  I,  IV, 


STRUCTURE    UF    MUSIC  95 

and  V  in  three  different  major  ke}-s,  and  V  and  VI  in  two 
different  minor  keys  (relationship  shown  by  formula  i).  As 
ev^ery  major  key  has  three  minor  triads  (ll,  III,  Vi)  and  every 
minor  key  has  two  minor  triads  (l,  iv),  therefore  every  minor 
triad  is  ll,  ill,  and  VI  in  three  different  major  keys,  and  I  and 
l\'  in  two  different  minor  keys  (relationship  shown  by  for- 
mula 2).  Also,  as  every  major  key  has  one  diminished  triad 
(vil°)  and  every  minor  key  has  two  diminished  triads  (ll° 
and  VII°),  therefore  every  diminished  triad  is  Vll°  in  one 
major  key,  and  11°  and  Vll°  in  two  different  minor  keys  (re- 
lationship shown  by  formula  3). 

10.  Therefore,  formulas  1,2,  and  3  cover  the  entire  ground 
of  modulation  by  common  (or  connecting)  triads  in  both  ma- 
jor and  minor  keys.  Thus,  if  any  major  triad  is  struck  in  any 
key  (major  or  minor)  it  may  be  regarded  as  belonging  also  to 
any  one  of  four  other  keys  related  as  shown  in  formula  i,  and 
if  any  minor  triad  is  struck  in  any  key  (major  or  minor)  it 
may  be  regarded  as  belonging  also  to  any  one  of  four  other 
keys  related  as  shown  in  formula  2,  and  if  any  diminished 
triad  is  struck  in  any  key  (major  or  minor)  it  maybe  regarded 
as  belonging  also  to  either  of  two  other  keys  related  as  shown 
in  formula  3,  the  modulation  depending  entirely  upon  the 
resolution  of  the  triad  in  each  case. 

11.  To  modulate  beyond  the  reach  of  any  one  of  the  for- 
mulas, we  may  reach  to  any  convenient  key  through  one  of 
the  formulas  (or  any  part  of  it),  then  move  to  another  triad  of 
same  key,  then,  with  the  formula  containing  this  new  triad, 
reach  (as  before)  to  another  convenient  key,  and  so  on  until 
the  desired  key  is  reached. 

12.  To  confirm  the  modulation  (if  not  already  confirmed), 
on  reaching  the  desired  key  we  should  resolve  to  a  distin- 
guishing chord  of  this  key.  The  distinguishing  chord  should 
include  the  7th  of  the  key  when  modulating  in  the  direction 
of  sharps  (or  the  relative  minor),  or  the  4th  of  the  key  (the 
7th  is  also  desirable,  as  it  is  the  leading  tone)  when   modulat- 


9^  MUSICOLOGY 

ing  in  the  direction  of  flats;  since  the  last  sharp  is  on  the  7th 
and  the  last  flat  is  on  the  4th.  The  YA^\  chord  (contain- 
ing both  4  and  7)  is  best  adapted  for  this  purpose. 

13.  Hence  the  rule:  A  key  is  usually  confirmed  by  some 
form  of  domifiant  harmony.  Other  chords  may  lead  to  or 
introduce  the  new  key. 

14.  The  V,  chord  is  a  very  restless  chord  and  calls  strongly 
for  the  Tonic  (I)  chord,  the  modulation  not  being  completed 
until  the  Tonic,  or  chord  of  home  feeling,  is  reached. 

15.  Modulation  by  Means  of  Connecting  Tones.  As 
already  "observed,  keys  are  related  to  each  other  in  propor- 
tion to  the  number  of  tones  they  have  in  common  ;  by  the 
same  principle,  chords  (whether  in  the  same  key  or  different 
keys)  are  also  related  in  proportion  to  the  number  of  tones 
in  common.  These  common  tones  are  naturally  connecting 
tones  in  passing  from  one  chord  to  another  (evidently,  the 
more  tones  in  common  the  less  motion  required"). 

16.  A  diatonic  chord  is  one  which  does  not  require  any 
change  in  the  natural  intervals  of  the  key  (involving  only  the 
sharps  or  flats  of  the  signature) ;  all  others  are  called  altered 
or  chromatic  chords. 

17.  There  are  only  four  kinds  of  diatonic  triads  in  a  key: 
the  major  triad  (major  3d  +  minor  3d),  the  minor  triad 
(minor  3d  +  major  3d),  the  dim.  triad  (minor  3d  +  minor 
3d),  and  the  aug.  triad  (major  3d  +  major  3d). 

18.  Observe  that  there  are  two  ways  of  changing  a  triad 
of  one  kind  into  another  kind — one  by  flats,  the  other  by 
sharps — since  moving  one  tone  one  half-step  in  one  direction 
or  moving  the  two  other  tones  one  half-step  in  the  other 
direction  will  have  the  same  effect  on  the  intervals  involved, 
but  of  course  the  modulation  will  be  in  opposite  directions. 

19.  Thus,  for  example,  we  may  change  the  major  triad  | 
into  a  minor  triad  either  by  flatting  the  3tl  or  sharping  the 
ist  and    5th,   as  b?  or  ^7.      In  either   case  we  have  changed 


STRUCTURE    OF    MUSIC 


97 


the  intervals  of  the  major  triad  (major  3d  +  minor  3d)  into 
the  intervals  of  the  minor  triad  (minor  3d  +  major  3d), 
(See  major  pattern.)  However,  neither  b?  nor  !  7  is  now  a 
diatonic  (but  a  chromatic)  chord  of  the  old  key,  but  is  a  dia- 
tonic chord  of  a  new  key.  As  they  are  now  minor  triads 
they  may  each  exist  (under  different  names)  in  five  different 
keys  related  as  shown  in  formula  2,  but  the  modulation  would 
most  naturally  be  to  the  one  nearest  related  to  the  key  from 
which  we  started.  See  key  table  and  observe  that  if  we 
start  from  ?  in  key  of  C  major,  then  b?  will  be  bl  (ll)  in  key 
of  F,  and  $  ?  will  be  H  (III)  in  key  of  E. 


rv  vii' 


Fig.  24. 


20.  Formulas  1,2,  and  3  may  be  combined,  as  in  Fig.  24, 
by  repeating  the  formulas  in  order,  thus  forming  an  unbroken 
chain.  The  ends  of  the  chain  may  be  lapped,  forming  a 
circle  (corresponding  to  the  key  circle).  A  copy  of  this  com- 
bination will  be  found  at  back  of  book,  which  may  be  cut  out 
and  applied  to  the  Combined  Key  Table  to  show  the  modu- 
lation effected  by  changing  any  triad  of  one  kind  into  a  triad 
of  any  other  kind. 

21.  Thus,  starting  with  the  formula  corresponding  to  the 
nature  (major,  minor,  or  diminished)  of  the  given  triad,  and 
placing  the  given  triad  in  line  with  the  key  we  start  from, 
then,  if  the  change  is  by  flats,  the  formula  to  the  left  corre- 
sponding to  the  nature  of  the  changed  triad  will  locate  all 
the  keys  containing  the  changed  triad.  If  the  change  is  by 
sharps,  the  formula  to  the  rigiit  corresponding  to  the  nature 
of  the  changed  triad  will  locate  all  the  keys  containing  the 
changed  triad.      Though  the  modulation  in  either  case   leads 


98  MUSICOLOGY 

first  to  the  key  nearest  related  to  the  starting  key  (as  differ- 
ing by  the  fewest  tones),  yet  the  changed  triad  may  be  re- 
solved into  any  one  of  the  different  keys  in  which  it  is  com- 
mon (as  already  shown  in  the  method  by  connecting  chords). 

22.  If  the  change  is  to  an  augmented  triad,  then  III'  to 
the  right  or  left  (as  the  case  may  be)  will  locate  the  new  key. 

23.  However,  the  principal  chords  used  in  modulation  by 
connecting  tones  are  the  Dominant  7th  (V,)  chord,  in  modu- 
lating in  major  keys  and  major  to  minor,  and  the  Diminished 
7th  (vii°)  chord,  in  modulating  in  minor  keys  and  minor  to 
major. 

24.  Modulation  with  the  Dominant  7th  Chord.  The  Dom- 
inant 7th  (V-)  chord  is  peculiarly  adapted  for  modulating  pur- 
poses for  several  reasons. 

25.  First,  the  V,  (  f  jchord  contains  the  three  key  limiting 
points:  7  (toward  the  left),  4  (toward  the  right),  and  5 
(toward  the  relative  minor).  See  key  table  (p.  46)  and  ob- 
serve that  we  cannot  move  from  7  of  any  key  to  the  left  (on 
same  degree  of  staff),  nor  from  4  to  the  right,  nor  from  5  to 
relative  minor,  without  chromatic  change  ;  but  all  four  tones 
of  the  chord  may  be  moved  to  the  tonic  minor  without 
change.  Hence  the  V.  chord  limits  the  key  in  all  directions 
except  toward  the  tonic  minor. 

26.  Second,  the  V^  chord  calls  strongly  for'  the  Tonic  chord 
(7  being  the  leading  tone  one  half-step  below  i,  4  being  one 
half-step  above  3,  and  5  being  a  connecting  tone). 

27.  Third,  there  is  no  other  chord  built  like  it  (major  3d, 
perfect  5th,  minor  7th),  so  that  there  is  no  other  chord  like 
it  in  the  same  key  (the  same  V,  chord  is  common  only  to  one 
major  key  and  its  tonic  minor — see  formula  i).  The  V, 
chord,  therefore,  has  a  peculiar  locating  power  and  always  calls 
for  a  certain  major  key  or  its  tonic  minor,  whichever  is  near- 
est related  to  tlie  key  from  which  we  are  modulating. 

28.  It  is  evident  that  u  e  can  form  the  interx-als   of   the  V, 


STRUCTURE    OF    MUSIC 


99 


chord  on  any  tone  of  any  key  by  using  the  necessary  sharps 
or  flats,  and  that  each  chord  thus  formed  will  be  the  diatonic 
V,  chord  of  some  key  and  will  call  for  that  key. 

29.  I,  and  IV,  differ  from  V,  only  by  the  7th  being  major 
instead   of  minor  (see    major  key  pattern),  therefore    I,  and 

IV,  can  be  changed  to  a  V,  either  by  flatting  the  7th  or  by 
sharping  the  ist,  3d,  and  5th. 

30.  II,,  III,,  and  VI,  differ  from  V,  only  by  the  3d  being 
minor  instead  of  major,  therefore  II,,  III,,  and  \i.,  can  be 
changed  to  a  V,  either  by  sharping  the  3d  or  by  flatting  the 
1st,  5th,  and  7th. 

31.  VII°,  differs  from  V,  by  the  3d  being  minor  and  the  5th 
being  diminished,  therefore  Vii°  can  be  changed  to  V,  either 
by  sharping  the  3d  and  5th  or  by  flatting  the  1st  and  7th. 

32.  Thus  we  can  form  the  V,  chord  in  twelve  different  ways 
in  any  one  key,  each  of  which  would  be  a  diatonic  V.  chord 
in  one  of  the  twelve  major  keys  and  its  tonic  minor,  and 
which  would  modulate  to  one  of  the  major  keys  except  where 
the  tonic  minor  comes  in  first  as  being  nearer  related.  (These 
chords  would  be  chromatic  chords  viewed  from  the  old  kev, 
but  diatonic  chords  viewed  from  the  new  key.) 

33.  We  may  begin  at  any  major  key,  but  the  key  of  C 
(being  without  sharp  or  flat)  is  the  most  convenient  for  pur- 
poses of  comparison.  Compare  the  following  results  by  re- 
ferring to  the  key  table  (p.  46). 

34.  Forming  the  V,  chord  on  i  by  flatting  the  7th,  thus, 
^,  we  have  the  V,  chord  of  the  key  of  F  or  /,  which  there- 
fore modulates  to   the    key    of    F    as    being    nearer   related. 

Sharping  the  1st,  3d,  and  5tli,  thus,  j^,  we  have  the  V.  chord 

o{  /^  or  F  ^,  and  modulation  is  to/|^  as  being  nearer  related. 

!j3  3 

35.  Forming  the  V,  chord  on  4.  thus,  I  or  jl,  we  have  the 

*       J* 

V,  chords  of  BI?  or  /'^  and  /;  or  B. 


lOO  MUSICOLOGY 

36.  Forming  the  V,  chord    on  2,  thus,  ^  or  ^l,  we  have 

3         b2 
the  V,  chords  of  G  or^,  and  G^  or  ^'■). 

2        b2 

37.  Forming  the  V^  chord  on  3,  thus,  4  or  ^4,  we  have  the 

V,  chords  of  a  or  A,  and  A  7  or  a'?. 

5  bs 

38.  Forming  the  V.  chord  on  6,  thus,  jf  or  "j,  we  have   the 

6  be 

V,  chords  of  D  or  ii,  and  D  7  or  ^/?. 

6  be 

39.  Forming  the  V,  chord  on  7,  thus,  j*  or   i,  we  have  the 

7  b- 

V,  chords  of  e  or  E,  and  E  !^  or  r  ?. 

40.  We  have  thus  modulated  to  twelve  difTerent  major  or 
minor  keys:  the  key  (major  or  minor)  given  first  in  each  case 
being  nearest  related  to  C. 

41.  If  we  had  taken  any  other  major  key  than  C  as  a  start- 
ing-point, we  would  have  obtained  similar  results,  the  only 
difference  being  that  we  would  have  had  to  use  naturals  to 
sharp  the  flatted  tones  in  flat  keys,  and  to  flat  the  sharped 
tones  in  sharp  keys. 

42.  Observe,  in  the  key  table,  that  a  minor  (relative  of  C 
major)  is  nearer  related  to  C  by  two  tones  (3d  and  6th)  than 
its  tonic  A  major.  Therefore,  all  the  minor  keys  to  the  right 
of  a  are  also  nearer  related  by  two  tones  (3d  and  6th)  than 
their  tonic  majors.  The  result  will  be  similar  if  we  start  from 
any  other  major  key  than  C.  Therefore,  in  modulating  with 
the  V,  chord  we  cannot  go  directly  beyond  the  nearest  two 
major  keys  to  the  right  of  the  starting  key.  after  which  the 
modulation  leads  into  the  minor  keys  (being  nearer  related 
than  their  tonic  majors);  but  toward  the  left  the  modulation 
leads  to  the  major  keys  (being  nearer  related  than  their  tonic 
minors).  In  cither  case  the  final  modulation  is  determined 
by  the  resolution  of  the  V,  chord,  which  may  resolve  to  the 
Tonic  chord  of  either  the  tonic  major  or  minor  key. 

43.  Another  method  of  modulating  with   the  V,  chord  is 


STRUCTURE    OF    MUSIC  lOI 

by  resolving  the  V,  chord  of  the  old  key  directly  into  the  V, 
chord  of  the  new  key  by  moving  the  tones  that  are  not  com- 
mon each  to  the  nearest  tone  of  the  new  V,  chord  and  using 
the  common  tones,  if  any,  as  connecting  tones.  Comparing 
the  V,  chords  of  the  different  keys  (see  key  table)  will  show 
the  moves  necessary  in  each  case. 

44.  We  may  also  modulate  by  making  a  chromatic  (half- 
step)  run  up  to  7  or  down  to  4  of  the  desired  key,  then  taking 
the  rest  of  the  V,  chord  of  that  key. 

45.  Observe  (see  Fig.  10,  p.  43)  that  sharping  the  root  (5) 
of  the  V,  chord  of  the  major  key  gives  the  Vil°  (Diminished 
7th)  chord  of  the  relative  minor  key.  Also  observe  (Fig.  1 1, 
p.  44)  that  flatting  the  7th  (6)  of  the  Vll°  chord  of  the  major 
key  gives  the  Vll°  chord  of  the  tonic  minor.  Therefore,  we 
may  modulate  to  any  minor  key  by  first  modulating  to  the 
V,  chord  of  its  relative  major,  then  sharping  the  root  (5) — 
by  making  both  moves  at  the  same  time  the  modulation 
would  be  direct;  or,  first  modulating  to  the  VII°  chord  of  its 
tonic  major,  then  flatting  the  7th  (6) — or  direct,  by  making 
both  moves  at  the  same  time.  The  two  methods  when  direct 
(in  one  move)  are  necessarily  the  same,  as  they  result  in  the 
same  chord — Diminished  7th  (vn,)  of  the  minor  key. 

46.  Modulation  with  the  Diminished  7th  (viJt)  Chord. 
This  chord  is  also  peculiarly  adapted  for  modulating  purposes 
for  several  reasons. 

47.  First,  it  is  only  found  as  a  diatonic  chord  in  minor 
keys  (on  the  7th),  and  therefore  calls  for  a  minor  key. 

48.  Second,  it  contains  all  four  of  the  key  limiting  points 
of  minor  keys   (see  pp.  47:   17-48:   18);    7   and  4   from   the 

,  tonic  view,  and  2  and  6  from  the  relative  view.  Observe, 
from  the  key  table,  that  we  cannot  move  (on  same  degree  of 
staff)  from  7  either  to  the  right  or  left,  nor  from  4  to  the 
right,  nor  from  2  to  the  left,  nor  from  6  to  the  right,  nor 
from  7  to  relative  major,  nor  from  6  to  tonic  major,  without 
chromatic  changes.     So  that  the  vn°   chord  limits  the  key  in 


102  MUSICOLOGY 

all  directions  (toward  the  right  by  7,  4,  and  6,  toward  the 
left  by  7  and  2,  toward  the  relative  major  by  7,  toward  the 
tonic  major  by  6). 

49.  Third,  it  can  easily  be  converted  into  a  V,  chord. 

50.  Fourth,  no  other  chord  is  built  like  it  (minor  3d,  dim. 
5th,  dim.  7th),  so  there  is  no  other  chord  like  it  in  the  same 
key. 

51.  Fifth,  three  of  its  members  are  capable  of  an  enhar- 
monic change,  each  change  converting  it  into  the  same  chord 
of  another  key.  (Modulation  by  means  of  enharmonic 
changes  is  called  eiihannoiiic  modulation.') 

52.  Comparing  any  minor  key  with  its  relative  and  tonic 
majors  (see  key  table),  we  will  see  that  the  VIT°  chord  of  any 
minor  key  can  be  changed  into  the  V,  chord  of  its  relative 
major  by  simply  lowering  the  root  (7)  one  half-step  (see  Fig. 
10,  p.  43),  or  into  the  Vll°  chord  of  its  tonic  major  Sy  rais- 
ing the  7th  (6)  one  half-step  (see  Fig.  11,  p.  44V  Thus  we 
have  a  simple  method  of  modulating  from  any  minor  key  to 
its  relative  and  tonic  majors,  and  vice  versa. 

53.  By  referring  to  the  minor  key  pattern  we  see  that  the 
VI  r^  chord  is  made  up  of  three  minor  3ds,  one  above  the  other, 
and  that  the  augmented  2d  between  6  and  7  (the  extremes  of 
the  chord,  harmonically  speaking)  is  equal  to  a  minor  3d  (U 
steps).  Therefore  the  VIl"  chord  is  practically  a  circle  of 
minor  3ds,  and  the  aug.  2d  may  take  the  place  of  any  one 
of  the  minor  3ds  without  affecting  the  tones,  though  causing 
a  refiguring  of  the  tones,  since  the  aug.  2d  must  come  between 
6  and  7,  and  involving  also  an  enharmonic  change  in  one  or 
two  of  its  members. 

54.  We  observe  also  (from  relative  minor  pattern")  that 
three  of  the  members  of  the  vii°^  chord  are  capable  of  an  en- 
harmonic change  (change  of  name  without  change  of  tone) ; 
thus,  Jt7  toh,  2  to  1?  3,  and  6  to  ^5.      Applying  these  changes 

6  6  6  55 

to  the  VI r,   chord,  we  would   have    *■     2.    ba-  '^"^1    |  (in  each 

^  J7     tti     bi  V 


STRUCTURE    OF    MUSIC  IO3 

case  changing  a  minor  3d  to  an  aug.  2d),  but  the  last  three 
will  have  to  be  refigured,  so  that    the   aug.  2d  will   come  be- 

6       4        2  3t7 

tvveen  6   and  7  ;    this  would   give   |,    ,,  ^,1,  and    ".      Thus  we 

87    be    b4  (a 

have  the  VIl^  chord  in  four  different  minor  keys  (made  up  of 
common  tones,  though  involving  different  accidentals  in 
writing  it  in  the  different  keys).  If  the  first  key  is  a  minor, 
the  second  will  be  ^  minor,  the  third  r>  minor,  and  the  fourth 
/If  minor.     (The  Vll,  of  tonic  minor  would  give  similar  results). 

55.  We  may  observe  that  every  third  minor  key  (see  minor 
key  table,  p.  46)  is  thus  related  through  the  common  Vli, 
chord,  and  also  that  these  same  keys  are  linked  together 
through  their  majors  (the  relative  major  of  one  being  the 
tonic  major  of  the  next,  etc.). 

56.  In  this  sense  the  Vll°  chord,  like  the  major,  minor, 
and  dim.  triads,  is  an  equivocal  chord  because  it  does  not 
point  to  any  particular  key,  since  it  may  resolve  into  the 
Tonic  chord  of  any  one  of  four  different  minor  keys  (the 
modulation  depending  on  the  resolution  in  each  case). 

57.  The  Vll°  may  evidently  be  regarded  as  the  V^  chord, 
with  root  omitted,  of  a  minor  key  (see  p.  72  :  22),  thus  being 
classed  as  dominant  harmony,  and  therefore  included  in  the 
rule  that  a  nciv  key  is  usually  co)ifirmcd  through  sonic  form  of 
the  Dominant. 

58.  Observe,  from  the  key  table,  that  in  resolving  theVll° 
chord  of  one  key  to  the  vil°  of  the  next  key  to  the  right,  we 
raise  each  tone  one  half-step  ;  or  to  the  vil°  of  the  next  key 
to  the  left,  we  lower  each  tone  one  half-step — there  being  no 
connecting  tone. 

59.  It  is  evident  that  the  intervals  of  the  Vli'^  chord  are 
the  same  in  its  several  positions  and  inversions,  since  the 
members  are  all  equidistant. 

60.  By  combining  the  several  methods  of  modulation  by  the 
Vll*^  chord,  we  may  modulate  from  any  key  to  any  other. 
Thus,  beginning  at  any  minor  key,  we  may  modulate  to  any 


I04  MUSICOLOGY 

minor  key  having  the  vif^  common  by  simply  resolving  it  to 
the  Tonic  chord  of  the  desired  key,  or  to  any  other  minor 
key  by  first  resolving  it  to  the  VII°  of  the  next  key  (right  or 
left),  then  resolving  to  the  Tonic  of  the  desired  key,  or  to  any 
major  key  by  changing  the  Vll°  chord  (before  resolving  to 
the  Tonic)  to  the  V^  of  its  relative  major  or  the  vii°  of  its 
tonic  major;  or,  beginning  at  any  major  key,  by  first  chang- 
ing the  vil°  of  the  tonic  major  or  the  V,  of  the  relative 
major  to  the  vil°  of  the  minor,  then  proceeding  as  above  in- 
dicated. We  make  the  modulation  abrupt  (direct),  in  any 
case,  by  making  the  different  moves  at  the  same  time. 

6i.  Beginning  at  the  major  key  of  F^,  we  may  take  its 
vii''.  chord  and  make  natural  the  7th  (6),  which  gives  the 
VII°  of  its  tonic  minor /if ,  then  Avith  the  common  Vll°  (by 
making  the  proper  enharmonic  changes)  we  may  step  over  to 
e\^  minor,  then  flatting  the  root  (7)  we  have  the  V,  of  its  rel- 
ative major  G^;  thus  we  have  modulated  from  F  |^  to  G^. 
But  these  keys  are  equivalent  (interchangeable)  keys ;  there- 
fore, by  using  the  extreme  limit  of  our  means  of  modulation, 
we  have  modulated  entirely  around  the  key  circle  to  the 
starting-point ;  or,  in  fact,  we  have  not  modulated  at  all,  but 
simply  made  the  enharmonic  change  and  moved  from  vii'i 
to  V,  of  equivalent  key. 

62.  It  is  evident  that  we  cannot  modulate  to  advantage, 
in  either  direction  or  by  any  means,  farther  than  half-way 
around  the  key  circle,  as  it  is  then  nearer  to  modulate  in  the 
opposite  direction  ;  and,  for  the  same  reason,  if  we  extend 
the  table  to  avoid  the  enharmonic  change,  we  cannot  modu- 
late to  advantage  farther  than  the  equivalent  of  half-way 
around  the  circle,  as  extending  the  table  is  merely  adding 
overlapping  keys  in  the  key  circle. 

63.  Modulation  with  the  Augmented  6th  Chord.  The 
Aj(ginentcd  6th  chord  is  a  chord  containing  the  interval  of  an 
aug.  6th  (5  steps).  It  is  used  in  three  forms,  called  Italian 
6th,  French  6th,  and  German  6th. 


STRUCTURE    OF    MUSIC  IO5 

64.  The  Italian  6th  chord  consists  of  a  major  3d  and  aug. 
6th  ;  thus,  ^  (in  minor  key).  The  French  6th  consists  of  a 
major  3d,  aug,  4th,  and  aug.  6th  (same  as  Italian  6th,  with 
aug.  4th  added) ;    thus,  f  (in  minor  key).      The    German   6th 

6 

consists  of  a  major   3d,  perfect    5th,  and   aug.    6th   (same   as 

Italian  6th,  with  perfect  5th  added);    thus,  ^  (in  minor   key). 

If  we  omit  the  5th  in  the  German  6th,  we  have  the  Italian 
6th  ;  or  if  we  change  the  5th  to  an  aug.  4th,  we  have  the 
French  6th. 

65.  The  thorough-bass  figuring  of  the  Italian  6th  is  j|6,  of 

the  French  6th  is  ^l,  of  the  German  6th  is  *§.      These  chords 
3 

are  only  used  in  direct  form. 

66.  The  Italian  6th  is  a  three-toned  chord,  but  in  forming 
four-part  harmony  the  3d  only  must  be  doubled. 

6y.  Observe  that  the  intervals  of  an  aug.  6th  and  minor 
7th  are  equal.  Therefore  the  German  6th  and  V,  chords  are 
composed  of  the  same  intervals  (as  to  measure),  and  one  can 
be  converted  into  the  other  by  enharmonically  changing  one 
member;  but  the  two  chords  resolve  differently — the  aug, 
6th  ascends,  while  the  minor  7th  descends, 

68.  The  Italian  6th,  French  6th,  and  German  6th,  being 
dissonant  chords,  require  resolution.  The  primary  resolu- 
tion of  each   is  the  same  (V,  with    root   doubled),  and   there- 

fore  (to  save  space)  may  be  treated  together.      Thus,    ?  2)   re- 

6 

solves  into  57  (minor  key).      It  will   be   seen   (see   minor   pat- 
5 

tern)  that  the  6th  (j|  4)  ascends  one  half-step  to  5,  and  the 
other  three  members  descend  each  one  half-step.  In  the  case 
of  the  Italian  6th — the  5th  (3)  being  omitted — the  3d  (i) 
may  either  descend  one  half-step  to  ||  7  or  ascend  one  step  to 
2  (or  if  the  3d  is  doubled,  it  may  take  both  parts).  In  the 
case  of  the  French  6th  the  4th  (2)  will  remain  stationary. 

69.  In  the  case  of  the  German  6th,  if  the  ist  and  5th  both 
descend   one   halt-step  at  the  same  time,  we  would  have  con- 


Io6  l\IUSICOLOGV 

secutive  5ths.  To  avoid  this  we  may  retard  the  resolution 
of  the  3d  and  5th  until  after  the  other  parts  resolve  P|,  f,  J  j, 
thus  forming  the  |  (2d  inversion)  of  the  minor  Tonic  between  ; 
or  the  5th  may  ascend  one  half-step  /  |,  *J,  jf  i,  forming  the  | 
of  the  major  Tonic  between;  or  we  may  preresolve  the  5th 
I  ^f'  ^1-  jj?  )•  forming  the  French  6th  between. 

70.  The  natural  seat  of  the  aug.  6th  {It.,  Fr. ,  or  Ger.) 
chord  is  on  the  6th  of  a  minor  key,  as  only  in  that  position 
is  it  in  the  same  key  with  its  chord  of  resolution,  as  will  be 
seen  by  trying  it  on  all  the  degrees  of  the  major  and  minor 
patterns. 

71.  The  aug.  6th  chord  is  often  used  merely  as  a  chro- 
matic chord,  but  when  used  as  a  means  of  modulation  it  nat- 
urally leads  to  that  minor  key  where  it  is  based  on  the  6th  of 
the  key  and  is  immediately  confirmed  by  its  chord  of  resolu- 
tion. (In  the  aug.  6th  chord  the  last  sharp  is  on  the  4th  of 
the  new  key  and  is  an  exception  to  the  rule  that  the  last 
sharp  is  always  on  the  7th  of  the  key.) 

72.  Since  the  German  6th  and  the  V,  chords  may  be  en- 
harmonically  changed  the  one  into  the  other,  they  may,  in 
a  sense,  be  regarded  as  a  common  chord  of  different  keys, 
the  modulation  being  determined  by  its  resolution.  Observe, 
from  the  key  table,  that  the  German  6th  chord  of  any  minor 
key,  as  (for  example)  a  minor,  is  composed  of  the  same  tones 
(enharmonically  changing  the  6th  to  a  7th)  as  the  V.  chord 
of  the  key  of  B  1?  major  (or  its  tonic  d  I'  minor).  Therefore, 
in  enharmonically  changing  a  German  6th  to  a  V,  chord,  we 
modulate  from  a  certain  minor  key  to  the  second  key  to  the 
left  of  its  relative  major,  or  conversely,  when  enharmonically 
changing  a  V,  to  a  German  6th  chord. 

Modulation  by  Inversion.     (Seep.  120.) 

Modulation  by  Imitation.     (Seep.  125.) 


STRUCTURE    OF    MUSIC  IO7 

SUMMARY    OF    MODULATION 

1.  The  diatonic  tones  of  any  key  are  its  natural  tones  un- 
affected by  accidentals  (involving  only  the  sharps  or  flats  in 
the  signature),  except  the  7th  in  minor  keys,  which  is  dia- 
tonic, though  marked  by  an  accidental. 

2.  If  a  diatonic  tone  of  a  key  is  chromatically  altered  by 
an  accidental,  it  is  called  a  chromatic  or  altered  tone.  A  tone 
may  be  chromatic  in  one  key  and  diatonic  in  another.  As 
already  observed  (p.  92:  11),  an  accidental  does  not  always 
produce  a  change  of  key,  as  it  may  sometimes  indicate  only 
a  chromatic  passing  tone  or  chord. 

3.  h.  diatonic  cJiord  (p.  96:  16)  is  one  that  contains  only 
diatonic  tones.  A  chromatic  chord  is  one  that  contains  one 
or  more  chromatic  tones,  A  chord  may  be  chromatic  in  one 
key  and  diatonic  in  another.  A  modulating  chord  contain- 
ing accidentals  is  a  chromatic  chord  as  viewed  from  the  old 
key,  but  a  diatonic  chord  as  viewed  from  the  new  key;  and 
modulation  is  only  the  natural  result  of  a  chromatic  chord 
calling  for  that  key  in  which  it  is  a  diatonic  or  natural  chord. 
If  a  chromatic  chord  does  not  follow  up  its  suggestion  of  a 
new  key,  but  returns  immediately  to  a  distinguishing  chord 
of  the  old  key,  it  should  be  regarded  merely  as  a  chromatic 
chord  of  the  old  key. 

4.  Sometimes  a  chromatic  chord  is  followed  by  another 
chromatic  chord,  the  two  suggesting  keys  in  opposite  direc- 
tions, in  which  case  they  counteract  each  other  and  no  mod- 
ulation is  produced. 

5.  We  see,  therefore,  that  an  accidental  does  not  always 
produce  modulation.  On  the  other  hand,  modulation  some- 
times takes  place  without  an  accidental  or  some  distance  in 
advance  of  the  accidental  (or  sign  of  the  new  key),  as  modu- 
lation takes  place  whenever  the  home-feeling  changes  to  a  new 
tonic,  whether  an  accidental  occurs  at  that  point  or  not ;  but  in 
most  cases  the  modulation  will  sooner  or  later  be  confirmed 
by  the  appearance  of  the  accidental.  (This  will  be  referred  to 
again  under  Modulation  by  "  Inversion,"  and  by  "Imitation.") 


I08  MUSICOLOGY 

6.  It  is  evident,  especially  between  any  two  closely  related 
keys  where  the  tones  are  nearly  all  common,  that  the  music 
may  continue  some  distance  on  chords  common  to  both  keys; 
the  music  being,  as  it  were,  on  a  balance  between  the  two 
keys,  so  that  a  slight  cause  or  suggestion  (or  even  a  slight 
mental  effort)  may  turn  the  balance  without  necessarily  in- 
volving an  accidental.  Therefore,  the  new  key  may  be  taken 
at  any  convenient  point  (when  the  music  is  thus  on  common 
ground),  as,  for  instance,  the  beginning  of  a  line  of  words,  or 
the  beginning  of  a  musical  phrase  or  figure,  or  any  other  nat- 
ural division  of  the  music,  or  after  a  rest,  or  when  the  arrange- 
ment of  tones  or  chords  seems  to  suggest  the  new  key. 

7.  There  may  be  three  stages  of  modulation — the  new 
key  being  first  suggested,  then  confirmed,  then  completed. 
The  suggestion  and  confirmation  may  or  may  not  take 
place  at  the  same  time.  We  accept  the  new  key  Avhen  first 
suggested,  but  the  suggestion  must  sooner  or  later  be  con- 
firmed, usually  by  some  form  of  Dominant  harmony  of  the 
new  key ;  we  are  still  not  at  home  in  the  new  key  till  we 
reach  the  Tonic,  or  chord  of  home-feeling,  thus  making  a 
complete  cadence  in  the  new  key. 

8.  If  a  modulation  is  very  short  and  incomplete  it  is  called 
a  transient  modulation  or  merely  a  digression  ;  but  if  a  com- 
plete cadence  (Dominant  followed  by  Tonic  harmony)  is  made 
in  the  new  key,  it  is  called  a  eadential  niodiilation. 

9.  Modulation  may  be  classified  as  follows: 

\  Xaiural  (to  a  closely  related  key). 
As  to  Distance  -^  Remote  (to  a  distant  key). 

.  ,,  j  Gradual  (passing  through  the  intermediate  keys). 

^    °       "^    '  I  Abrupt  (stepping  over  the  intermediate  keys). 

{Diatonic  (by  means  of  common  chords). 
Chromatic  (by  means  of  chromatic  chords). 
Enharmonic  (by  means  of  enharmonic  changes). 

(  Digressive  (when  short  and  incomplete). 
As  to  Completeness -!  Cadential  (when   making  a  complete  cadence  in 
(      the  new  key). 


STRUCTURE    OF    MUSIC  IO9 

10.  Duration,  location,  the  form  and  position  of  the  Dom- 
inant and  Tonic  chords,  the  place  in  rhythm,  etc., — all  have 
a  bearing  on  the  importance  of  a  modulation. 

1 1.  The  different  methods  of  modulation  already  explained 
bear  only  on  the  question  of  "  How  to  modulate."  "  When 
to  modulate"  is  an  equally  important  question. 

12.  The  art  of  modulating  does  not  consist  merely  in  the 
ability  to  ramble  through  various  keys,  but  consists  in  know- 
ing when,  as  well  as  how,  to  modulate. 

13.  The  purpose  of  modulation  is  to  enlarge  the  means  of 
musical  expression  by  placing  the  combined  harmonies  of  all 
the  keys  at  our  disposal  instead  of  confining  us  to  the  limits 
of  one  key. 

14.  We  may  enter  a  new  key  whenever  the  old  key  proves 
insuflficient  to  produce  the  musical  effect  we  wish ;  also,  to 
avoid  too  much  uniformity  (especially  in  the  longer  pieces  of 
music),  and  to  secure  variety  and  contrast.  Just  as  disso- 
nances heighten  by  contrast  the  effect  of  consonances,  so 
modulation  heightens  by  contrast  the  sense  of  key  tonality ; 
but  either  should  be  used  only  when  it  gives  a  sense  of  justi- 
fication in  the  effect  produced.  Modulation  should  thus  be 
the  result  of  necessity  rather  than  a  display  of  skill. 

15.  Our  sense  of  symmetry  and  proper  balance  requires 
that  the  music  should  end  in  the  same  key  in  which  it  begins ; 
otherwise,  it  would  seem  incomplete. 

TRANSPOSITION 

I.  As  already  seen,  modulation  consists  in  passing  from 
one  key  to  another  during  the  course  of  the  music,  thus  in- 
volving only  accidentals.  On  the  other  hand,  transposition 
consists  in  moving  the  entire  music  bodily  into  another  key, 
thus  involving  a  new  signature.  However,  if  the  transposi- 
tion be  by  the  interval  of  an  octave,  the  signature  (and  there- 
fore the  key)  will  not  be  affected,  since  the  scale  is  not  changed. 
But    if   the   transposition   be  by  any  other  interval  than    the 


no  MUSICOLOGY 

octave,  the  music  will   evidently  be   in    a   different   scale,  or 
key,  requirini^  the  corresponding  signature. 

2.  Music  may  be  transposed,  or  moved  bodily,  to  any  key 
exactly  as  the  key  pattern  (Chart  I.)  may  be  moved  bodily  to 
any  key  (the  music  being  made  up  of  key  tones).  And  as 
moving  the  key  pattern  does  not  affect  the  relation  of  the  key 
tones  to  each  other,  so  transposition  does  not  affect  the 
interrelationship  of  the  tones  of  the  music.  Transposition, 
therefore,  is  merely  a  change  of  pitch  of  the  entire  music. 

3.  Any  modulations  of  the  music  before  it  is  transposed 
will,  after  it  is  transposed,  be  similarly  related  to  the  new 
key  as  formerly  to  the  old,  and  involving  therefore  the  same 
accidentals. 

4.  Writing  the  transposed  music  will  evidently  involve 
both  a  new  signature  and  the  raising  or  lowering  (as  the  case 
may  be)  of  each  note  on  the  staff  a  distance  equal  to  the  in- 
terval of  transposition. 

5.  Transposition  does  not  affect  the  reading  of  music  nor 
the  figuring  of  chords,  as  each  note  has  the  same  position 
(and  therefore  the  same  key  syllable  name  and  numeral)  in 
the  new  key  as  formerly  in  the  old,  the  entire  music  remain- 
ing intact  though  in  a  new  key. 

6.  Music  may  be  transposed  in  the  singing  or  playing  of  it 
without  rewriting  it.  As  the  reading  of  the  music  is  not 
changed,  it  is  only  necessary,  therefore,  to  apply  the  key  sylla- 
bles or  numerals  with  the  key-note  pitched  to  the  desired 
key. 

7.  When  several  persons  sing  together  it  is  necessary  that 
they  use  the  same  key;  but  when  a  person  sings  alone  the 
music  may  be  pitched  in  any  key  that  best  suits  his  voice, 
regardless  of  the  key  in  which  it  is  written.  In  singing,  the 
key  is  merely  a  question  of  pitching  the  voice  at  the  start, 
the  music  naturally  following  in  the  key  thus  established. 

8.  In  a  similar  manner,  playing  on  the  keyboard  in  a  dif- 
ferent kev  from   that  in  which    the   music   is   written   consists 


STRUCTURE    OF    MUSIC  III 

merely  in  moving  the  key-note  to  the  right  or  left  (as  the  case 
may  be)  and  playing  the  music  accordingly  with  reference  to 
the  new  key-note.  This  will  involve  both  reading  and  play- 
ing the  music  with  reference  to  the  syllable  or  numeral  name 
of  each  note  regardless  of  its  degree  on  the  staff.  A  study 
of  Chart  II.  (at  back  of  book),  by  moving  the  key  pattern 
to  the  different  keys  and  remembering  that  the  pattern  repre- 
sents the  music  to  be  transposed,  will  make  the  principle 
clear. 

9.  Each  position  of  the  pattern  shows  the  white  and  black 
keys  used  in  playing  in  that  key.  In  playing  in  the  key  of 
C,  only  white  keys  are  used,  and  the  only  difference  in  play- 
ing in  other  keys  is  that  certain  black  keys  are  substituted 
for  the  white  keys  to  the  right  or  left  (according  as  the  music 
is  in  flats  or  sharps),  and  therefore  requires  a  mental  note  of 
what  keys  are  thus  substituted.  Also,  E  and  F,  and  B  and 
C  are,  in  certain  keys  (with  six  or  seven  sharps  or  flats  in  sig- 
nature), substituted  for  each  other,  as  explained  on  p.  34:  21. 

10.  We  may  also  observe  that  each  interval  includes  a  cer- 
tain number  of  finger-bars  (taking  the  white  and  black  suc- 
cessively), and  that  the  same  interval  includes  the  same  num- 
ber of  finger-bars  at  any  place  on  the  keyboard,  and  there- 
fore the  same  hand-span  spans  the  same  interval  at  any  place 
on  the  keyboard.  This  involves  the  development  of  what  is 
called  the  "sense  of  location,"  by  which  one  may  uncon- 
sciously measure  the  intervals  on  the  keyboard  by  hand-spans, 
just  as  in  singing  he  unconsciously  measures  the  intervals  by 
their  key  tonality  or  mental  effect. 

11.  Transposition  to  the  tonic  minor  involves  merely  the 
flatting  of  the  3d  and  6th  of  the  key,  but  transposition  to 
the  relative  minor  involves  lowering  the  entire  music  a  minor 
3d  and  sharping  the  /th  of  the  key. 


112  MUSICULUGY 

COUNTERPOINT 

1.  Counterpoint  (meaning  the  point  opposite  or  counter j 
refers  to  writing  notes  (formerly  called  points)  opposite  or 
counter  to  others,  or  the  art  of  writing  music  in  parts. 

2.  Harmony  is  the  accompaniment  of  melody  or  tune 
with  chords,  while  counterpoint  is  a  combination  of  melodies, 
or  an  accompaniment  of  melody  with  melody,  the  chief 
aim  in  counterpoint  being  to  give  a  melodic  flow  to  each  part. 

3.  Counterpoint  is  much  older  than  harmony,  and  there- 
fore precedes  harmony  from  the  view  of  priority ;  however, 
the  same  general  principles  are  involved  in  both,  which  are 
most  naturally  treated  under  harmony,  for  which  reason  har- 
mony is  usually  treated  first. 

4.  The  earliest  part-music  consisted  of  melody  accompa- 
nied in  the  octave,  5th,  or  4th.  In  time,  the  singers  began 
to  vary  the  monotony  of  these  intervals  by  adding  (extempo- 
raneously) what  are  now  called  passing  notes.  It  was  also 
discovered,  about  this  time,  that  several  independent  melo- 
dies could  be  sung  together  with  good  effect  by  adapting 
them  to  each  other  in  rhythm  and  pitch  and  making  slight 
changes  where  the  dissonances  were  too  apparent.  The  pop- 
ularity of  these  new  methods  naturally  led  musical  composers 
to  combine  the  principles  involved  by  adding  melodic  parts 
to  their  principal  melodies  (also  to  existing  melodies),  hence 
the  origin  of  counterpoint. 

5.  At  first  the  whole  interest  was  centered  in  the  melodic 
movement  of  the  parts,  the  harmony  being  merely  the  inci- 
dent result  of  avoiding  dissonances.  On  tiie  other  hand,  in 
modern  harmony  the  whole  interest  (aside  from  the  single 
melody)  is  centered  in  the  harmonic  combination  of  tones. 

6.  Naturally  the  highest  development  of  music  is  in  the 
combination  of  these  two  elements  of  interest,  or  the  har- 
monic combination  of  melodies.  Counterpoint  must  there- 
fore conform  both  to  the  requirements  of  melody  and  to  the 
requirements  of  harmony. 


STRUCTURE    OF    MUSIC 


113 


7.    Counterpoint  may  be  outlined  as  follows: 


3 

as 
6  > 


■  Analysis 


I    - 


Species 


Subject  (or  Canius  Firmus). 

Added  part  or  parts  (the  counterpoint  proper). 

1st  species  (one  note  of  the  counterpoint  to  each  note 

of  the  C.  F. — note  against  note). 
2d  species  (two  notes  of  the  counterpoint  to  each  note 

of  the  C.  F  — two  against  one). 
3d  species  (four  notes  of  the  counterpoint  to  each  note 

of  the  C.  F. — four  against  one). 
Syncopated  (each  note  of  the  counterpoiiU  syncopated 

with  each  note  of  the  C.  F.). 
Florid  (a   mixture   or  combination   of    the   foregoing 

species). 

(Inversion  means  the  upper  part  placed  below  or  the  lower  part  above.) 

[Double  (two  parts  which    also    make    correct    music 

when  inverted). 
Triple  (three  parts   which    also    make    correct    music 

in  any  order). 
Quadruple  (four  parts  which  also  make  correct  music 

in  any  order). 
Manifold  (any  greater  number  of  invertible  parts). 


Inversion  in  the  octave. 
Inversion  in  the  gth  (octave  +  2d). 
Inversion  in  the  loth  (octave  +  3d). 
Inversion  in  the  nth  (octave  +  4th). 
Inversion  in  the  12th  (octave  +  5th). 
Inversion  in  the  13th  (octave  +  6th). 
Inversion  in  the  14th  (octave  +  7th). 
Inversion  in  the  15th  (double  octave). 


Classes 


Interval 

of 

.  Inversion 


8.  In  Simple  Counterpoint  a  melody — called  the  subject 
ox  cant  us  fir  urns — is  taken,  to  which  one  or  more  melodic  or 
flowing  parts  are  added,  the  part  or  parts  thus  added  being 
called  the  counterpoint . 

9.  The  cantus firuius  maybe  in  any  part,  and  therefore  the 
counterpoint  may  be  written  above  the  cantus  firuius,  or  be- 
low, or  both. 

10.  Simple    counterpoint    includes  five    species.      In    the 


114 


MUSIC()LO(]Y 


first  species  one  note  is  written  in   the  counterpoint  to  each 
note  of  ihcitnitus Ji}-iiins;  thus, 


j,  called  note  against 


note.      In   the   second    species,  two  notes   are   written  in  the 

r I  I  --■ 

counterpoint  to  each  note  of  the  r^/////^-_/?>w//^,  thus,  P — ^ — ^^\j 

^-~~ — zy ; 

called  two  against  one.  (In  triple  time  three  notes  are  writ- 
ten against  one — the  notes  of  the  r^r ///■// .vyfrw/zjr  being  dotted.) 
In  the    third   species,  four   notes  are  written  in   the    counter- 

point  to  each  note  of  the  caiitj/s  Jiniius,  thus,  f    J    J    #  ■  J    \ 

called  four  against  one.  (In  triple  time  six  notes  are  written 
against  one — the  notes  of  the  cantns  firinus  being  dotted.) 
In  syncopated  counterpoint  two  notes  are  written  to  each 
note  of  the  cantiis  firmus,  but  syncopated,  or  tied  over,  thus, 

■i — ^/H d — H        In   Jiorid    counterpomt 


^ 


any  or  all  of  the  foregoing  species  are  combined.  When 
more  than  one  part  is  added,  they  ma\-  be  of  different 
species. 

11,  In  counterpoint,  as  in  harmony,  each  combination  of 
tones  (except  passing  tones)  represents  some  definite  chord 
(complete  or  incomplete).  All  intervals  are  classed  as  con- 
cords or  discords,  the  concords  being  the  octave  and  per- 
fect 5th  (called  perfect  concords),  and  major  and  minor  3ds 
and  6ths  (called  imperfect  concords).  All  other  intervals  are 
treated  as  discords,  including  the  perfect  4th  when  between 
the    lowest    and    one     of    the    upper     parts    (see    foot-note, 

p.  83). 

12.  Contrapuntal  Rules.  Contrai)untal  music  was  origi- 
nally for  vocal  performance;  and  the  rules  of  counterpoint 
were  for  the  purpose  of  making  the  i)arts  easih'  singable,  as 
well  as  to  avoid  disagreeable  effects.  Progressions  that  were 
found  to  be  awkward  or  difificult  for  singers  to  take  in  con- 
nection with  other  voices,  or  combinations  that  produced  dis- 


STRUCTURE    OF    MUSIC  II5 

agreeable  effects,  were  naturally  forbidden — hence  the  origin 
of  the  rules. 

13.  In  modern  counterpoint  the  rules  are  not  so  rigidly 
observed,  as  the  ears  of  modern  singers  are  more  accustomed 
to  dissonant  chords  ;  while  in  instrumental  counterpoint  the 
players  are  not  dependent  on  the  guidance  of  the  ear.  How- 
ever, the  influence  of  the  rules  is  always  beneficial. 

14.  (i)  In  strict  counterpoint  of  the  first  species  {note  against 
note)  otily  consonant  combinations  should  be  used,  as  consonant 
combinations  are  easier  to  sing  than  dissonant  ones,  as  ivell  as 
more  agreeable.  When  two  or  more  notes  are  written  against 
one,  a  consonant  combination  may  change  conjunctly  (with- 
out skip)  into  a  dissonance,  as  a  dissonance  thus  taken  is  not 
difficult  to  intone. 

15.  (2)  Accented  pulses  should,  as  a  rule,  be  consonant  and 
so  far  as  possible  complete  chords,  while  unaccoitcd pulses  may 
be  either  consonant  or  disso)iant.  In  syncopated  counterpoint 
the  first  note  of  each  syncopation  must  be  consonant,  the  second 
note  being  eitJier  consonant  or  dissonant.  The  rules  for  the 
resolution  of  dissonances  (pp.  83,  84)  are  applied. 

16.  (3)  Skips.  Consonant  intervals  are  free  to  move  by  skips. 
Dissonant  intervals  should,  as  a  rule,  be  approached  and  quit- 
ted ivithout  skips.  A  skip  from  a  discord  on  an  unaccented 
pulse  to  a  concord — called  a  changing  tone — is  sometimes 
used.  Concordant  skips  (3ds,  6ths,  perfect  4ths,  perfect  5ths, 
and  octaves)  are  freely  used.  Discordant  skips  (/ths,  aug- 
mented and  diminished  intervals)  should  be  avoided.  A 
succession  of  wide  skips  (or  more  than  two  skips — even  of  a 
3d)  in  the  same  direction  should  be  avoided.  After  using 
three  or  four  notes  alphabetically  a  skip,  even  of  a  3d,  in  the 
same  direction  should  be  avoided  ;  but  the  skip  may  be  made 
at  the  beginning  of  the  passage. 

17.  (4)  The  Tritone  (aug.  4th  =  3  steps).  This  interval 
occurs  naturally  between  the  4th  and  7th  of  major  and  minor 
keys  (see  major  and  minor  patterns).   It  is  not  only  to  be  avoid- 


Il6  MUSICOLOGY 

ed  as  a  skip,  but  also  hcti'.'coi  parts  of  successive  chords,  thus, 
|?y— #^-^^;  or  hetu'ccn  parts  of  chords  on  successive  accents; 


or  bettveoi  the  first  and  last  notes  of  an  ascending  or  descend- 


ing passage,  thus. 


^^ 


1 8.  (5)  Consecutives.  Similar  perfect  concords  {octaves  and 
perfect  jths)  should  not  occur  successively  nor  on  successive 
accents,  especially  betzveen  outside  parts.  The  same  applies, 
though  less  rigidly,  to  hidden  consecutives  (p.  8  i  :  24).  Tivo  suc- 
cessive Jiiajor  jds  {unless  taken  by  a  half -step,  as  on  the  jth  and 
6th  of  minor  keys— see  mitior  pattern)  should  be  avoided. 
Successive  jds  and  6ths  should,  so  far  as  possible,  be  alternate- 
ly major  and  minor. 

19.  (6)  False  Relations.  False  relations  should  be  avoided 
(p.  81  :  26,  27). 

20.  (7)  Doubling.  Double  major  jds  should  be  avoided,  es- 
pecially on  the  accents.  The  same  applies  less  rigidly  to  the 
6ths  and  minor  jds.  Doubled  leading  notes  should  especially 
be  avoided. 

21.  (8)  Unison.  The  u)iison  may  occur  in  the  first  and  last 
chords  and  occasionally  on  unaccented  pulses,  but  should  be  used 
sparingly. 

22.  (9)  Crossing.  'The  parts  may  occasionally  cross  for  the 
sake  of  a  more  melodious  flow,  but  should  not  eross  on  the 
accent. 

23.  (10)  Pedal.  J\dal  passages  (p.  88)  are  occasionally 
used. 

24.  (11)  Sequence.  Sequences  (p.  go)  are  always  effective, 
and  may  be  freely  used. 

25.  (12)  Modulation.  Modulation  to  closely  related  keys  is 
not  restricted. 

26.  (13)  Motion.  Oblique  and  contrary  motion  (^p.  81  :  28) 
should  be  used  as  much  as  possible,  thus  giving  more  individu- 


STRUCTURE    OF    MUSIC  II/ 

ality  to  the  parts  and  lessening  the  liability  to  faults.  All  the 
parts  slumld  rarely  move  in  the  same  direction  at  the  same 
time. 

27.  F"aults  are  less  observed  between  inside  parts  or  be- 
tween inside  and  outside  parts,  also  on  unaccented  pulses, 
and  also  as  the  parts  increase  in  number.  On  the  other 
hand,  increased  number  of  parts  necessitates  increased  free- 
dom of  movement  to  secure  the  necessary  flow  to  all  the 
parts.  Therefore,  rules  are  less  rigidly  applied  as  the  parts 
increase  in  number,  or  between  inside  parts,  or  between  in- 
side and  outside  parts,  or  on  unaccented  pulses. 

28.  Double  Counterpoint  consists  of  two  invertible  and 
equally  important  parts. 

29.  The  following  table  shows  the  intervals  of  inversion, 
and  the  intervals  within  each  and  their  inversions. 

TABLE    OF    INVERSIONS 

Inversion  in  octave    j     i     2     3     4     5     6     7     3— intervals 
(    S     7     6     5     4     3     2     I — inversions 

Inversion  in  the  Qth   |    1    "     £    4     5    ^     7    ^     9-intervals 
(98765432     I  —  inversions 

Inversion  in  the  loth  I     '     1     ^     4     5     6^8     9  lo-intervals 
(  10     9     S     765432     I — inversions 

Inversion  in  the  nth  .^_[     2     3^    4    ^    6    T    S^    9  i^  n-intervals 
(11   10     9S7     6     5432     I — inversions 

,             .       .,            ,\l234s67Snioii    12 — intervals 
Inversion  in  the  I2th  ■         _     "     _  __         

(12111098765432     I — inversions 

,            .       .      ,           ,(12345678     9   10  II    12   13 — intervals 
Inversion  in  the  13th  -  __  _  

(  13   12   II    10     9     S     7     6     5     4     3     2     I — inversions 


3 


T     .   .   ,     ,  \  I  2  3  4  5  6  7  8  9  10  II  12  13  14 — interval 

Inversion  in  the  14th -' „__  _ 

/  14  13  12  II  10  9  8  7  6  5  4  3  2  I — invers. 

nversi.jn  inthe  15th  i  i   2  3  4  5  6  7  S  9  10  11  12  13  14  15  —int. 

(  15  14  13  12  II  10  9  8  7  6  5  4  3  2  I  — inv. 


Il8  MUSICOLOGY 

30.  Observe  that  any  interval  added  to  its  inversion  is  one 
greater  (the  central  note  on  which  the  inversion  turns  being 
counted  twice)  than  the  interval  in  which  the  inversion  is  to 
be  made.  This  suggests  how  the  inversion  of  any  interval 
may  readily  be  found. 

31.  In  the  above  table  the  dissonant  intervals  are  marked 
with  a  stroke  over,  thus,  — .  Observe  tliat  in  the  octave  all  con- 
sonant intervals  invert  into  consonant  intervals  and  dissonant 
intervals  into  dissonant  intervals,  except  4  and  5  ;  the  same 
in  the  lOtJi  without  any  exceptions ;  the  same  in  the  12th, 
except  6  and  7 ;  and  that  the  /J//^  (double  octave)  corre- 
sponds to  the  octave  (iiths  and  I2ths  being  compound  4ths 
and  5ths).  These  are  the  intervals  of  inversion  most  used. 
It  will  be  seen  that  the  others  invert  so  contrary  as  to  give 
less  satisfactory  results. 

32.  Two  parts  intended  for  inversion  should  not  at  any 
point  exceed  the  compass  of  the  interval  of  inversion  ;  other- 
wise, the  parts  will  cross  when  inverted  (^the  portion  in  ex- 
cess remaining  on  the  same  side  as  before  inversion,  and  there- 
fore not  inverted  but  merely  contracted).  Likewise,  the  two 
parts  intended  for  inversion  should  not  cross  at  any  point,  as 
the  reverse  of  the  above  will  happen  when  inverted  (the  por- 
tion crossed  expanding  instead  of  inverting).  Therefore,  the 
greater  the  interval  of  inversion,  the  greater  the  range  allowed 
to  the  parts.  It  is  evident  that  when  parts  are  inverted  as  a 
whole,  each  interval  between  corresponding  notes  is  inverted 
according  to  the  interval  of  inversion. 

33.  Special  attention  must  be  given  to  those  intervals  that 
invert  contrary  (as  regards  consonance),  the  other  intervals 
practically  taking  care  of  themselves  (if  forming  correct  pro- 
gressions, the  inversions  naturally  forming  correct  progres- 
sions also).  Consonant  intervals  that  invert  into  dissonant 
intervals  must  therefore  be  treated  as  dissonant  as  regards 
preparation  anil  resolution,  the  treatment,  however,  being  in 
the  lower  part,  which  becomes  the  dissonant   part   when  in- 


STRUCTURE    OF    MUSIC 


119 


verted.  Thus,  in  the  octave  the  perfect  5th,  though  conso- 
nant, inverts  into  the  perfect  4th,  which  is  dissonant,  and 
requires  resolution  and  usually  preparation,  especially  on  the 
accent;  the  perfect  5th  must  therefore  conform  to  the  same 
requirements  (in  the  lower  part)  in  order  to  form  correct  pro- 
gression when  inverted.  The  same  applies  to  inversion  in 
the  1 2th  as  regards  the  6th  and  7th  ;  however,  Dominant  /ths 
and  Diminished  /ths  do  not  usually  require  preparation  (see 
p.  82:30). 

34.  It  should  be  observed  in  the  lotJi  that  consecutive  3ds 
and  6ths  must  be  avoided,  as  they  become  consecutive  octaves 
and  5ths  in  the  inversion.  Also,  if  these  intervals  are  ap- 
proached by  similar  motion,  objectionable  hidden  octaves  and 
5ths  are  apt  to  occur  in  the  inversion.  It  is  apparent,  there- 
fore, that  oblique  and  contrary  motion  must  generally  be 
employed. 

35.  In  the  i2tJi,  3ds  invert  into  loths  and  vice  versa;  these 
intervals,  therefore,  may  freely  be  used  even  in  similar  motion  ; 
but  for  the  other  intervalsoblique 
and  contrary  motion  will  gen- 
erally be  found  necessary. 

36.  In  the  other  intervals  of 
inversion  (9th,  iith,  13th,  and 
14th,  which  are  but  little  used) 
it  is  evident  that  conjunct  move- 
ment (without  skip)  will  generally 
be  necessary  so  that  every  dis- 
cord may  be  approached  and 
quitted  conjunctly. 

37.  To  more  closely  analyze 
the  intervals  and  their  inver- 
sions, see  Fig.  25.  Each  dia- 
gram represents  a  section  of  the 
major  key  pattern,  in  which  the  major  intervals  are  measured 
from  each  end  (the  natural  intervals  upward  from  i  of  major  pat- 


(12  til) 

rl3 ^1-| 

(lOti^) 

11^2 

10 — 1- 

10 — 
-  3 
-9—4 

(8^e) 

-9—2 

r8 1-1 

-8—3 

-8 — 5- 

f1 

7 — 4 

7 

-       2 

- 

-       6- 

G 

-6—5 

6 

3 

- 

7- 

-5—4- 

-5— G 

-5 — 8- 

"4—5" 

-4 — ;- 

-3 — 8 

-4 — 9 
-3 

3 

-       6 

- 

-      10- 

2 

-2—9 

-2—11- 

■I 8- 

-l-^« 

-1—12- 

Fig.  25. 


I20  MUSICOLOCV 

tern   arc    taken     as    the  standard    and    called    major;     see    p. 
56:2). 

38.  Observe  that  in  the  octave  perfect  intervals  invert  in- 
to perfect  intervals,  major  into  minor,  and  vice  versa,  and 
diminished  into  augmented,  and  I'ice  versa. 

39.  Observe  that  in  the  lotJi,  major  intervals  invert  into 
major  intervals  (including  the  perfect  intervals,  which  are 
sometimes  called  major),  minor  intervals  (including  the 
diminished  perfects,  which  are  sometimes  called  minor) 
invert  into  augmented  intervals,  and  vice  versa.  The 
other  diminished  intervals,  being  equivalent  by  enharmonic 
change  to  major  or  perfect  intervals,  invert  accordingly  ;  thus, 
the  diminished  7th,  being  equivalent  to  a  major  6th,  inverts 
into  a  perfect  5th  (otherwise  a  double-augmented  4th,  which 
is  not  a  classified  interval). 

40.  Observe  that  in  the  J2tJi,  3ds,  6ths,  /ths,  and  lOths 
invert  as  in  the  octave  (major  into  minor,  and  vice  versa,  aug- 
mented into  diminished,  and  vice  versa),  while  the  other  in- 
tervals invert  as  in  the  lotli. 

41.  Inversion  in  the  i^th  (double  octave)  is  practically  the 
same  as  in  the  octave. 

42.  The  other  intervals  of  inversion  may  be  analyzed  in  a 
similar  manner. 

43.  It  is  evident  that  inversion  may  sometimes  involve 
augmented  and  diminished  intervals,  which  are  usually  chro- 
matic. The  available  augmented  and  diminished  intervals 
(p.  57  :  II)  are  the  aug.  2d,  aug.  4th,  aug.  5th,  and  aug.  6th, 
and  the  dim.  5th  and  dim.  7th.  The  aug.  6th  cannot  be 
used  in  the  octave,  as  it  inverts  into  the  dim.  3d.  Of  course 
these  intervals  are  dissonant,  and  usually  require  treatment, 
and  also  involve  accidentals  which  tend  to  induce  modulation. 

44.  Modulation  by  Inversion.  If  in\ersion  in  the  octave 
takes  place  by  the  upper  part  being  placed  an  octave  lower, 
or  the  lower  part  an  octave  higher,  or  both  parts  an  octave 
in  opposite  directions,  no    modulation  occurs,  as  the  scale  is 


STRUCTURE    OF    MUSIC  12  1 

not  changed  ;  but  if  both  parts  arc  moved  in  opposite  direc- 
tions by  steps  that  involve  a  new  scale,  modulation  will  be 
effected  to  the  key  corresponding  to  the  new  scale.  Thus, 
if  the  lower  part  is  placed  a  perfect  5th  higher  and  the  upper 
part  a  perfect  4th  lower,  they  will  evidently  both  be  in  the 
scale,  or  key,  of  the  dominant;  or  if  the  lower  part  is  placed 
a  perfect  4th  higher  and  the  upper  part  a  perfect  5th  lower, 
they  will  be  in  the  scale,  or  key,  of  the  sub-dominant.  Modula- 
tion may  thus  be  effected  to  any  key  by  simply  moving  each 
part  bodily  to  the  scale  of  that  key  (moving  the  parts  in  such 
direction  as  will  produce  inversion). 

45.  Inversion  in  the  iot]i  can  only  be  effected  in  four 
ways:  first,  the  upper  part  may  be  placed  a  lOth  lower;  sec- 
oud,  the  lower  part  may  be  placed  a  loth  higher;  third,  the 
upper  part  may  be  placed  an  octave  lower,  and  the  lower  part 
a  3d  higher;  fourth,  the  lower  part  may  be  placed  an  octave 
higher,  and  the  upper  part  a  3d  lower.  Of  course  the  mod- 
ulation will  not  be  through  the  part  moving  an  octave  ;  but 
the  part  moving  a  3d  or  loth  (octave  +  3d)  will  evidently  be 
in  a  new  scale  a  major  3d  above  or  below  the  old  key,  accord- 
ing as  the  modulating  part  is  placed  above  or  below.  If  the 
old  key  is  C,  the  new  key  will  be  E  or  A  7  (as  the  case  may  be). 

46.  If  the  inversion  were  by  any  other  than  the  four  ways 
mentioned,  both  parts  would  modulate,  but  to  different  keys. 
Thus,  if  the  lower  part  is  placed  a  perfect  5th  higher  and  the 
upper  part  a  major  6th  lower,  then  one  part  would  be  in  the 
key  a  perfect  5th  above,  and  the  other  in  the  key  a  major  6th 
below,  the  old  key.  If  the  old  key  is  C,  then  one  part  would 
be  in  the  key  of  G  and  the  other  in  the  key  of  E  k;  but  the 
music  cannot  be  in  two  keys  at  the  same  time. 

47.  Modulation  by  inversion  in  the  12th  corresponds  in 
every  point  to  that  in  the  loth  by  simply  substituting  the  per- 
fect 5th  for  the  major  3d.  The  modulation  will  be  to  the 
key  of  the   dominant   (perfect    5th  above)  or   sub-dominant 


122  MUSICOLOGY 

(perfect    5tli    below),    according    as    the    modulating    part    is 
placed  above  or  below. 

48.  Modulation  by  inversion  in  the  isth  corresponds  to 
that  in  the  octave. 

49.  Modulation  by  inversion  in  the  ^///,  /////,  ijtJi,  and 
i^th  is  similar  to  that  in  the  loth  and  I2t]i  (differing  only  in 
the  modulating  step). 

50.  In  any  case,  the  modulation  is  caused  by  one  or  both 
parts  moving  by  some  other  interval  than  an  octave.  Though 
the  modulation  may  be  only  in  one  part  so  far  as  the  acci- 
dentals involved  are  concerned,  yet,  of  course,  both  or  all  the 
parts  are  included  in  the  modulation,  since  the  music  cannot 
be  in  different  keys  at  the  same  time.  The  modulation  will 
naturally  begin  with  and  continue  through  the  entire  section 
of  the  music  thus  inverted,  and  may  therefore  begin  some 
distance  in  advance  of  the  accidental,  or  sign  of  modula- 
tion. 

51.  If  modulation  occurs  in  the  original  parts  (before  in- 
version), then,  of  course,  the  modulation  by  inversion  will  be 
reckoned  (during  the  continuance  of  that  modulation  only) 
from  that  key  instead  of  from  the  original  or  signature 
key. 

52.  It  sometimes  occurs  (where  the  keys  are  closely  related) 
that  the  tone  in  which  the  keys  differ  (and  therefore  the  acci- 
dental) does  not  happen  to  be  used,  yet  still  modulation  may 
result  through  the  tendency  of  the  part  inverted  to  retain  and 
carry  with  it  its  key  individuality  (if  strong),  since  the  inter- 
relationship of  its  tones  remains  intact.  Thus  modulation 
sometimes  occurs  without  accidentals.  Modulation  is  effected 
whenever  the  feeling  of  modulation  is  produced. 

53.  Counterpoint  Invertible  in  Various  Intervals.  Coun- 
terpoint in  the  octave  may  also  invert  in  the  lotJi,  12th,  or 
75///,  since  its  compass  is  within  the  others.  For  the  same 
reason,  counterpoint  in  the  loth  may  also  invert  in  the  12th 
or  iStJt ;  and   counterpoint  in  the  i2tJi  may  also  invert  in  the 


STRUCTURE    OF    MUSIC  1 25 

i^th.      In  each  case,  however,  the  parts  should  be  constructed 
with  reference  to  each  inversion. 

54.  Counterpoint  may  also  be  made  invertible  in  different 
intervals  by  adding  3ds.  This  consists  in  duplicating  one  or 
both  parts  in  the  3d  above  or  below,  thus  giving  three  or 
four  parts.  If  in  counterpoint  in  the  octave  3ds  be  added 
above  the  upper  part  or  below  the  lower  part,  the  new  part 
thus  added  will  be  in  the  loth  with  the  other  outside  part, 
and  the  counterpoint  is  thus  invertible  in  both  the  octave  and 
lotJi.  If  3ds  are  added  both  above  the  upper  part  and  below 
the  lower  part,  then  the  inner  parts  will  be  in  the  octave,  the 
inner  and  outer  parts  in  the  lotJi,  and  the  outer  parts  in  the 
i2tJL.  If  the  original  parts  invert  in  the  I2t]i  and  3ds  added 
both  below  the  upper  and  above  the  lower  part,  the  result 
will  be  the  same. 

55.  In  applying  this  method  of  added  3ds,  the  following 
additional  rules  will  be  necessary : 

(i)   Use  dissonances  only  as  passing  notes. 

(2)  Use    consonant   intervals   alternately   as  much   as 

possible. 

(3)  Use  only  oblique  or  contrary  motion. 

56.  Triple  and  Quadruple  Counterpoint.  These  consist  of 
three  or  four  distinct  parts,  each  standing  to  each  in  the  rela- 
tion of  a  double  counterpoint  in  the  octave. 

57.  Triple  counterpoint  will  give  six  different  combinations, 
and  quadruple  counterpoint  will  give  twenty-four  different 
combinations,  according  to  the  principle  of  Permutation. 

IMITATION 

1.  Counterpoint  is  made  up  largely  of  Imitation. 

2.  Partial  hnitation  is  where  a  part  of  a  preceding  melody 
is  imitated.  Canonical  Imitation  (see  Canon,  p.  128)  is  where 
the  preceding  melody  is  imitated  throughout. 

3.  Imitation  may  take  place  in   the  unison  or  at  any  other 


124 


MUSTCOLor.Y 


interval,  in  any  part  or   number  of   parts,  aiul   at    any  place. 
The  part  imitated  may  be  called  the  model. 

4,  Imitation  may  be  outlined  thus: 

f  Strict. 
Free. 

Retrograde. 
Imitation.  \  By  Contrary  Motion. 
By  Augmentation. 
By  Diminution. 
By  Reversed  Accents. 

5.  In  Strict  Imitation  the  intervals  of  the  model  are  not 
changed — half-step  answering  to  half-step,  whole  step  to 
whole  step,  and  therefore  major  to  major,  and  minor  to 
minor.  This  is  the  usual  character  of  imitations  in  the 
unison  or  octave,  since  the  scale  is  not  changed. 

6,  In  Free  Imitation  some  of  the  intervals  of  the  model 
are  changed — the  imitation  being  by  similar  degrees  of  the 
staff  regardless  of  the  half-steps,  and  therefore  major  often 
answering  to  minor,  and  vice  versa.  This  is  the  usual  char- 
acter of  imitations  in  any  other  interval  than  the  unison  or 
octave,  since  the  scale  is  changed. 

7.  In  Retrograde  Imitation  the  model  is  imitated  backward, 
_j      from  end  to  beginning  (strict  or  free). 

8.  In  Imitation  by  Contrary  Motion  the  model  is  imi- 
tated upside  down  (strict  or  free).  If  strict,  the  imita- 
tion must  be  in  accordance  with  the  pattern  shown 
in  Fig.  26,  which  represents  a  section  of  the  major  key 
pattern  (and  may  be  extended  into  the  full  pattern). 

9.  Observe  that  the  reverse  scale  begins  a  3d  above 
the  key-note  (i)  of  the  direct  scale,  and  also  that  the 
intervals  of  both  direct  and  reverse  scales  are  the 
same.  Therefore,  if  the  imitation  corresponds  in  the 
reverse  scale  to  the  model  in  the  direct  scale,  it  will 
be  strict  (half-step  answering  to  half-step,  and  whole 

step   to  whole  step).      Of  course,  this  pattern,  like  the  key 
pattern,  may  be  set  to  any  key. 


3- 

— I- 

•2- 

-3- 

-1- 

-3 

•7- 

-4 

6- 

-5 

5- 

-6 

■4- 
-3- 

— 1- 

2- 

-2 

1— 

-* 

Fig.  2G. 


STRUCTURE    OF    MUSIC 


125 


f« — h 


4 — 2^ 
3 


-3 — 4 

-1—5^ 


-2 — 

4 — 5-" 

Fig.  27. 


10.  Imitation  by  contrary  motion  in  minor  keys  is  best 
made  by  the  pattern  shown  in  Fig.  27,  which  represents 
a  section  of  the  minor  key  pattern  (and  may  be  ex- 
tended into  the  full  pattern).  It  will  be  seen  that 
when  the  3d  of  either  scale  is  used,  the  imitation 
ceases  to  be  strict,  or  else  involves  a  chromatic  tone. 

11.  In  Iinitatioii  by  Augmentation  the  time  value 
of  each  note  of  the  model  is  increased  or  augmented 
(usually  doubled). 

12.  In  Iinitatioji  by  Diminntio)i  the  time  value  of 
each  note  of  the  model  is  diminished  (usually  one- 
half). 

13.  In  Imitation  by  Reversed  Accents  the  accents 
are  reversed  (unaccented  for  accented,  and  vice  versa) 
by  the  imitation  entering  on  an  opposite  phase  of  the 
measure  from  that  on  which  the  model  entered. 

14.  Modulation  by  Imitation.  If  the  imitation  takes  place 
in  the  unison  or  octave  it  is  evident  that  no  modulation  will 
be  involved,  since  the  scale  is  not  changed  ;  or  if  the  imita- 
tion be  free,  it  may  take  place  at  any  other  interval  without 
involving  modulation  (the  imitation  being  by  similar  degrees 
of  the  staff  regardless  of  the  half-steps) ;  but  if  the  imitation 
be  strict  and  at  any  interval  involving  a  new  scale,  modula- 
tion will  naturally  result.  The  interval  between  the  old  and 
new  key  letters  will  correspond  to  the  interval  at  which  the 
imitation  takes  place  ;  thus,  if  the  imitation  takes  place  at  the 
interval  of  a  perfect  5th  above  or  a  perfect  4th  below,  the 
modulation  will  be  to  the  key  of  the  dominant,  etc. 

15.  If  the  imitation  be  by  contrary  motion  in  accordance 
with  the  pattern  shown  in  Fig.  26,  no  modulation  will  be  in- 
volved, since  the  intervals  of  the  reverse  scale  correspond  to 
the  intervals  of  the  diatonic  scale  and  therefore  in  the  same 
key.  Otherwise  modulation  may  occur,  since  any  other  re- 
verse scale  would  be  in  some  other  key.  In  any  case,  3  of 
reverse  scale  will  be  on  the  key-note  as  in  pattern. 


126  MUSICOLOGY 

16.  In  minor  keys  if  the  imitation  by  contrary  motion  be 
in  accordance  with  pattern  shown  in  Fig.  27,  no  modulation 
will  be  involved,  for  same  reason  as  given  above  (3  being  free 
or  involving  a  chromatic  tone).  Otherwise  modulation  may 
occur.  In  any  case,  5  of  reverse  scale  will  be  on  key-note  as 
in  pattern. 

17.  Of  course,  in  any  case,  the  modulation  is  usually  indi- 
cated by  the  accidental  following.  The  modulation  naturally 
begins,  however,  with  the  imitation  and  therefore  frequently 
some  distance  in  advance  of  the  accidental.  It  sometimes 
happens  (when  the  keys  are  closely  related)  that  the  acciden- 
tal involved  in  the  new  key  is  not  required,  yet  modulation  may 
result  through  the  tendency  of  the  model  to  retain  and  carry 
with  it  its  key  individuality  (if  strong  and  the  imitation  strict). 

CONTRAPUNTAL    MUSIC 

1.  Any  combination  of  melodies  may  be  regarded  as  con- 
trapuntal music ;  but  the  most  important  contrapuntal  forms 
of  musical  composition  are  the  Fugue  SiWd  the  Ca?ion.  Points 
of  contrapuntal  imitation  are  frequently  found  in  other  music, 
especially  in  choruses. 

2.  The  Fugue  is  a  composition  in  which  a  musical  phrase 
called  the  Subject  is  given  out  by  one  part  and  imitated  by  the 
other  parts  in  succession,  thus: 

(  Soprano Answer  \ 

}  Alto Subject Counter-subject        f 

")  Tenor Answer Counter-subject Free  Counterpoint  f 

(  Bass Subject Counter-subject Free  Counterpoint Free  Counterpoint  ) 

(Any  part  may  begin  the  Fugue.) 

3.  The  A?is2c>er  is  the  subject  imitated  in  the  key  of  the 
dominant,  either  above  or  below.  The  Counter-subject  is  the 
continuation  of  the  subject  or  answer  during  the  imitation  in 
the  next  part. 

4.  The  counter-subject  need  not  appear  in  the  voice  which 
enters  last.  After  the  counter-subject  each  part  is  continued 
by  adding  free  counterpoint.  The  subject  and  counter-sub- 
ject should  be  in  double  counterpoint  so  that  they  may  be 


STRUCTURE    OF    MUSIC 


127 


inverted,  since  they  are  to  be  interwoven  in  all  possible  com- 
binations in  the  subsequent  treatment. 

5.  The  subject,  answer,  and  counter-subject  are  called  the 
Exposition,  as  they  form  the  subject-matter  from  which  the 
fugue  is  to  be  developed.  The  structural  form  of  the  fugue 
may  be  outlined  thus: 


I,   The  Exposition 


f  Subject. 
-  Answer. 
[  Counter-subject. 


Fugue  Form  -j 


Episode. 


2.  The  Contrapuntal  r>                  •            j       •      . 

I  '1  Repercussion  and  episode. 

I  Development        "1  c.     .. 

'-  ^                  I  Stretto. 

[  Final  episode. 

6.  The  Episodes  are  the  free  modulations  which  connect  the 
parts  and  lead  from  one  key  to  the  next. 

7.  The  Repercussion  consists  of  new  combinations  of  the  sub- 
ject and  answer  in  different  keys ;  each  combination  being 
followed  by  an  episode  leading  into  another  key. 

8.  The  Stretto  consists  of  combinations  in  which  the  sub- 
ject and  answer  overlap.  Where  there  are  several  strettos, 
they  should  come  in  the  order  of  their  closeness,  the  closest 
coming  last.  Sometimes  the  repercussion  is  made  up  of 
strettos. 

9.  The  principal  kinds  of  fugues  may  be  classified  thus: 

i  Two-voiced  fugues. 
As  to  number  of  voices       <  Three-voiced  fugues. 
(  Etc. 


As  to  number  of  subjects 


As  to  kind  of  imitation 


As  to  character  of  subject 


Single  fugues. 
Double  fugues. 
Etc. 

Augmented  fugues. 
Diminished  fugues. 
Inverted  fugues. 

Diatonic  fugues. 
Chromatic  fugues. 

A     .  1  .       .         .      \  Strict  fugues. 

As  to  general  treatment      \  ^ 

(  Free  fugues. 


128  MUSICULOGY 

10.  The  Canon  is  a  composition  in  which  the  parts  enter 
one  after  the  other,  each  imitating  the  melody  of  the  first 
part  (called  the  subject)  throughout. 

1 1.  An  imitating  part  may  enter  at  any  point,  and  at  any 
interval,  up  or  down.  The  difTerent  kinds  of  imitation  may 
be  employed  and  the  different  parts  may  employ  different 
kinds  of  imitation  ;  or  some  of  the  parts  may  be  in  Canon,  and 
the  other  parts  free.  There  are  also  canons  with  more  than 
one  subject. 

.  12.  Canons  are  usually  described  by  figures  showing  the 
number  of  parts  and  subjects,  the  first  figure  showing  the 
number  of  parts,  or  voices,  and  the  second  figure  showing  the 
number  of  subjects,  thus:  Canon  2  in  1,3  in  i,  4  in  2, 
etc. 

13.  K  Finite  Canon  is  one  in  which  each  part  is  silent  after 
completing  the  melody,  or  subject;  or  one  which  ends  with  a 
regular  close  like  other  compositions. 

14.  An  Infinite  Canon  is  one  in  which  each  part  begins 
again  after  completing  the  subject,  the  canon  thus  being  with- 
out end. 

15.  A  Circular  Canon  is  one  which  modulates  from  one 
key  to  the  next  around  the  key  circle,  the  subject  recom- 
mencing each  time  in  the  key  a  4th  or  5th  higher  or  lower; 
or  the  subject  may  recommence  a  tone  higlicr  or  lower,  thus 
modulating  by  alternate  kc)'s.  The  conclusion  of  the  sub- 
ject should  lead  naturally  each  time  into  the  new  key.  The 
circular  canon  is  also  endless. 

16.  An  Open  Canon  is  one  in  which  the  different  parts  are 
written  out  in  full. 

17.  A  Close  Canon  is  one  in  which  the  principal  part  only 
is  written  out,  and  the  number  of  parts  and  their  places  of 
entrance  are  indicated  by  the  sign  Jj. 

18.  The  Enigma  (or  riddle)  Canon  is  a  kind  of  musical 
problem  in  which  the  places  of  entrance  of  the  succeeding 
parts  are  not  indicated,  but  left  to  be  solved. 


STRUCTURE    OF    MUSIC 


129 


19.    The  canon  may  be  classified  as  follows: 


As  to  interval 
of  imitation 


Canons  in  unison. 
Canons  in  2d. 
Canons  in  3d. 
Canons  in  4th. 
Etc. 


Mixed  Canons 


I 

f  Canons  2  in  i. 
As  to  number  of  |  Canons  3  in  i. 

parts  and  subjects   j  Canons  4  in  2. 
I  Etc. 


\  the  parts  imitating  at 
(      different  intervals. 


As  to  strictness  of 
imitation 


As  to  style  of 
imitation 


As  to  limitations 


As  to  score 


I  Strict  canons  (see  Strict  Imitation). 
1  Free  canons  (see  Free  Imitation). 


I  Canons  in  similar  motion. 

!  Canons  in  contrary  motion  (see  pp.  124:0  ;  125:10). 

Canons  in  augmentation  (see  p.  125:11). 
j  Canons  in  diminution  (see  p.  12512). 
[Retrograde  canons  (see  Retrograde  Imitation). 

i  Finite  canons  (limited). 
-I  Infinite  canons  (endless). 
(  Circular  canons  (endless). 

(  Open  canons. 
I  Close  canons. 


Enigma  canons. 

20.  In  the  fugue  the  subject  is  sometimes  only  a  short 
musical  phrase.  In  the  canon  the  subject  is  an  entire 
melody. 


130  MUSICOLOGY 


MELODY 


1.  Music  is  divided  into  three  general  classes:  MonopJionic 
(single  sounding),  one-part  music  (melody)  ;  Polyphonic  {vad.v\y 
sounding),  combined  melodies  (counterpoint) ;  Harvionic 
(united  sounding),  accompanied  melody  (harmony). 

2.  AhmopJionic  music  (melody)  is  the  original  form  of  all 
music.  The  ancients,  though  sometimes  having  choirs  of 
many  voices,  sang  only  in  unison  (or  naturally  in  octaves, 
when  men  and  women  sang  together).  Melody  is  still  the 
prevailing  music  of  the  Turks,  Greeks,  and  most  oriental 
nations,  who  seem  to  have  a  distaste  iox  part  music. 

3.  Polyphonic  music  (counterpoint)  dates  back  about  seven 
hundred  years,  when  the  early  attempts  at  part  music  began 
to  assume  shape. 

4.  Harmonic  music  (harmony)  is  based  on  principles  of 
acoustics.  It  is  also,  in  a  sense,  a  gradual  outgrowth  of 
counterpoint,  and  dates  back  only  about  two  hundred  years 
as  a  recognized  distinct  system. 

5.  Melody  is  a  succession  of   tones,    while  harmony  is  a 

1  9 

concord  of  tones.      V^'^^W-^  m    -|~T~a--]     At    i  we     have  a 


concord  of  tones,  or  tones  heard  simultaneously  in  harmony ; 
while  at  2  we  have  the  same  tones,  but  heard  successively  in 
melody.  The  first  is  called  a  chord;  the  second,  a  melodic 
figure;  and  just  as  harmony  is  made  up  of  chords,  so  melody 
is  made  up  of  figures. 

6.  A  chord  and  a  figure  composed  of  the  same  notes  con- 
tain, in  a  sense,  the  same  musical  idea;  but  the  figure  con- 
veys the  additional  idea  of  motion,  while  the  chord  conveys 
the  idea  of  rest,  or  completeness  in  itself. 

7.  The  i)rinciples  of  harmony  as  applied  to  chords,  natu- 
rally, in  a  general  sense,  apply  also  to  figures ;    for  harmonic 


I 


STRUCTURE    OF    MUSIC  I3I 

combinations  naturally  make  melodic  successions.  In  this 
sense  a  figure  is  a  resolved  chord  and  melody  is  resolved  har- 
mony. But,  as  already  observed,  the  figure  conveys  the  ad- 
ditional idea  of  motion,  so  that  it  has  a  melodic  element  com- 
bined with  its  harmonic  clement ;  the  harmonic  connection  be- 
tween the  successive  notes  is  in  proportion  to  the  consonance 
of  the  interval  between,  while  the  melodic  connection  is  in 
proportion  to  the  closeness  of  the  interval  between. 

8.  The  melodic  importance  of  the  2d  is  due  to  its  being 
the  shortest  melodic  step.  It  is  freely  used  in  melody,  since 
the  notes  forming  it  do  not  sound  simultaneously.  It  is  thus 
often  used  as  a  passing  note  between  the  harmonically  con- 
nected tones  to  increase  the  smoothness  of  the  melodic  flow. 
It  is  the  only  diatonic  step  with  a  purely  melodic  value  (be- 
ing harmonically  dissonant) ;  the  other  diatonic  steps  used 
have  both  a  harmonic  and  melodic  value. 

9.  Melodic  Progression  involves  relationship  between  the 
consecutive  notes  (grouped  into  figures),  just  as  harmonic  pro- 
gression involves  relationship  between  the  consecutive  chords, 
and  just  as  speech  involves  grammatical  relationship  between 
the  different  words. 

10.  Melody  is  more  than  a  mere  succession  of  related  tones. 
For  a  succession  of  tones  to  be  melody  it  must  have  melodic 
flow  and  rhythmir  movement.  Melody,  therefore,  is  a  melo- 
dious, rhythmical  succession  of  tones. 

1 1.  l.^h.Q  figJire  is  the  smallest  complete  rhythmical  division 
of  a  melody.  The  figure,  therefore,  has  a  rhythmic  element 
in  combination  with  its  harmonic  and  melodic  elements.  The 
figure  usually  covers  one  or  two  measures,  called,  respectively, 
simple  and  compound  figures.  The  figure  may  begin  in  one 
measure  and  end  in  another,  in  which  case  the  melody  begins 
with  an  incomplete  measure. 

12.  Development.  A  figure  may  also  be  defined  as  a  series 
of  notes  containing  a  musical  idea,  and  which  may  be  taken 
as  a  subject,  or  theme,  for  development.      When  thus  used 


132  MUSICOLOGV 

the  figure  is  called  a  theme,  and   the  treatment   involved   in 
development  is  called  tJuniatic  trcatmoit. 

13.  The  principal  methods  of  thematic  treatment  may  be 
outlined  as  follows : 

Transposition  (placing  the  figure  at  a  higher  or  lower  pitch). 

Expansion  (expanding  one  or  more  intervals  of  the  figure). 

Contraction  (contracting  one  or  more  intervals).. 

Augmentation  (increasing  the  time  value  of  each  note). 

Diminution  (diminishing  the  time  value  of  each  note). 

Repetition  (repeating  fragments  or  certain  notes  of  the  figure). 

Omission  (omitting  fragments  or  notes).   ' 

Reversion  (the  figure  written  backward  from  end  to  beginning). 

Contrary  Motion  (the  figure  written  upside  down). 

Change  of  Order  (irregular  change  of  the  order  of  the  notes  of  the  figure). 

Rhythmic  Change  (change  of  the  rhythm  of  the  figure). 

Ornamentation  (with  various  passing  notes,  turns,  trills,  etc.). 

Simplification  (reverse  of  ornamentation). 

Combination  (any  two  or  more  of  the  above  combined). 

Some  of  these  methods  are  much  used,  and  others  but 
little  used. 

14.  Thematic  treatment  modifies  the  character  of  a  figure, 
or  theme,  without  entirely  destroying  its  individuality.  Of 
course,  if  a  figure  is  so  changed  as  to  lose  its  individuality  it 
becomes  a  new  figure.  The  figure  may  be  recognized  through 
any  one  or  more  of  its  three  elements  (harmonic,  melodic, 
and  rhythmic) ;  but  if  all  three  are  changed  there  is  no  means 
of  recognition,  and  the  figure  loses  its  individuality. 

15.  By  means  of  thematic  treatment  an  entire  melody  is 
often  developed  from  a  single  figure,  but  more  usually  a 
number  of  figures,  or  themes,  are  used:  the  first,  being  most 
prominent,  is  called  the  principal  theme ;  and  the  others, 
secondary  themes.  The  greater  the  number  of  themes,  the 
greater  the  variety  and  contrast ;  but  too  many  themes  tend 
to  destroy  the  unity  of  the  melody.  A  principal  idea  must 
pervade  the  entire  melody  ;  otherwise,  it  will  have  no  definite 
meaning;  just  as  a  literary  composition  without  a  principal 
subject,    or    theme,    would    ha\c    no    tlefiiiite    meaning,    and 


STRUCTURE    OF    MUSIC  1 33 

therefore  lack  interest  by  being  difficult  to  understand.  In 
either  case,  the  unity  or  oneness  of  the  composition  is 
through  a  principal  idea  pervading  the  whole.  On  the  other 
hand,  without  variety  and  contrast  the  composition  would 
lack  interest  by  being  monotonous. 

16.  In  melody  variety  is  largely  attained  by  thematic 
treatment  (development)  of  the  same  theme,  just  as  in  literary 
compositions  variety  is  largely  attained  by  presenting  the 
same  theme  in  various  lights  (also  thematic  treatment) ;  but 
in  either  case,  the  sense  of  the  principal  theme  is  often  most 
clearly  brought  out  by  using  contrasting  themes. 

LINGUISTIC   CHARACTER   OF    MUSIC 

1.  Music  should  not  be  looked  upon  merely  as  a  pleasing 
combination  of  tones.  Properly  understood,  music  is  a  lan- 
guage through  which  ideas  and  thoughts,  sentiment  and  emo- 
tion, are  expressed.  It  differs  from  spoken  language  in  that 
it  is  natural  and  universal  (in  the  form  of  melody),  while 
spoken  language  is  artificial  and  local. 

2.  Musical  perception  is  to  some  extent  a  natural  gift  pos- 
sessed by  all,  though  in  different  degrees,  and  which  is  highly 
susceptible  of  cultivation. 

3.  The  basis  of  spoken  language  is  the  alphabet ;  the  basis 
of  musical  language  is  the  music  scale. 

4.  Just  as  spoken  language  is  made  up  of  letters  combined 
into  words,  words  into  phrases,  phrases  into  clauses,  clauses 
into  sentences,  sentences  into  paragraphs,  and  paragraphs 
into  compositions,  so  musical  language  is  made  up  of  tones 
combined  into  figures  (or  chords  in  harmony),  figures  into 
phrases,  phrases  into  sections,  sections  into  periods,  periods 
into  movements,  and  movements  into  the  higher  forms  of 
musical  composition. 

5.  Melody  (being  regulated  by  rhythm)  may  be  called  the 
poetry  of  music,  and  the  recitative  (musical  declamation),  its 
prose. 


134 


MUSICOLOGY 


6.  Melodic  motion  imitates  mental  motion.  Thoughts  are 
mental  motions.  Our  thoughts  may  move  slow  or  fast ;  they 
may  revel  in  pleasant  fancies  or  wander  through  sad  memo- 
ries ;  they  may  be  aimless  or  energetic,  restless  or  deter- 
mined, calm  or  excited.  Melodic  motion  is  capable  of  ex- 
pressing by  imitation  every  species  of  mental  motion  in  the 
most  delicate  and  exact  manner. 

7.  Melody,  therefore,  is  more  than  merely  a  melodious, 
rhythmical  succession  of  tones ;  it  is  sentiment  and  emotion 
expressed  in  tones  instead  of  words. 


THE  PERIOD 

1.  The  Period  is  a  musical  sentence.  It  consists  of  several 
phrases  so  related  as  to  produce  the  sense  of  completeness. 

2.  The  Phrase  is  a  musical  expression  having  a  well 
marked  repose.  A  Passage  is  a  series  of  figures  having  no 
well  marked  repose. 

3.  Rhythmic  Structure.  The /rr/(3</ is  a  rhythmical  whole 
which,  in  its  regular  form,  is  divided  rhythmically  into  halves 
called  sections ;  the  sections,  rhythmically  into  halves  called 
phrases;  and  the  phrases,  rhythmically  into  halves  called  fig- 
ures, designs,  or  motives,  thus: 

Period 


Section 


Section 


Figure  A     Figure    A      Figure    A  Figure   A       Figure      A      Figure      A         Figure  A   Figure    / 

Phrase  A  Phrase  A  Phrase  A  Phrase  / 


Fig.  38. 


4.  The  figure  may  sometimes  be  divided  into  rhythmical 
fragments  called  germs.  (The  term  "rhythm"  in  its  strict 
sense  includes  not  only  the  smaller  rhythmical  divisions,  but 
also  the  larger  rhythmical  combinations.) 


STRUCTURE    OF    MUSIC  1 35 

5.  It  will  be  seen  that  the  above  period  contains  eight 
measures.  Evidently,  if  we  double  the  time  value  of  each 
note  without  changing  the  time  value  of  the  measure,  we 
would  double  the  number  of  measures,  and  vice  versa.  Also, 
if  we  divide  the  time  value  of  each  measure  (f  into  |,  or  §  into 
%,  etc.,  thus  dividing  each  measure  into  two)  without  chang- 
ing the  time  value  of  the  notes,  we  would  double  the  num- 
ber of  measures,  and  vice  versa.  The  character  and  number 
of  measures  depend  largely  upon  the  character  of  the 
figure. 

6.  The  regular  form  of  the  period  contains  four,  eight,  or 
sixteen  measures,  most  usually  eight.  When  incomplete 
measures  occur  at  the  beginning  and  end  of  the  period  they 
count  together  as  one  measure. 

7.  The  regular  form  of  the  period  may  be  contracted  or 
expanded  in  various  ways :  ist,  the  period  may  be  cut  short 
by  rests,  in  which  case  the  measure  (or  measures)  without 
notes  is  counted  as  part  of  the  period  ;  2d,  the  two  sections 
of  the  period  may  overlap,  the  last  measure  of  the  first  sec- 
tion being  also  the  first  measure  of  the  second  (periods  may 
also  overlap  in  the  same  way) ;  jd,  the  two  sections  may  be 
connected  by  a  short  passage  leading  from  one  to  the  other 
(periods  may  also  be  connected  in  the  same  way);  4th,  the 
first  two  measures  may  be  repeated  ;  ^th,  the  last  two  meas- 
ures may  be  repeated ;  6th,  single  figures  may  be  repeated  ; 
yth,  a  short  conclusion  may  be  added  ;  Sth,  a  short  introduc- 
tion may  be  prefixed  ;  etc.  However,  introductions,  conclu- 
sions, and  connecting  passages  are  not  strictly  a  part  of  the 
period. 

8.  The  rhythmical  division  of  the  period  sometimes  varies 
more  or  less  from  the  regular  form  given. 

9.  Thematic  Structure.  Since  melody  is  made  up  of  peri- 
ods just  as  spoken  language  is  made  up  of  sentences,  there- 
fore what  was  said  on  the  development  of  melody  by  means 
of  thematic  treatment  applies  to  the  structure  of  periods. 


136  MUSICOLOGY 

10.  The  character  of  a  tone  is  its  mental  effect  or  tonality. 
These  combined  make  up  the  character  of  the  theme  (figure). 
These,  in  turn  combined,  make  up  the  character  of  the 
phrase,  and  so  on  up.  So  that  each  combination  in  turn  is, 
in  a  sense,  a  larger  theme  with  a  character  made  up  of  the 
combined  characters  of  its  component  parts.  Ascending  series 
of  tones  express  exaltation  ;  descending  series  of  tones  ex- 
press relaxation.  A  series  which  both  ascends  and  descends 
is  vague,  having  no  special  character;  but  if  on  the  whole 
ascending  it  partakes  of  the  character  of  an  ascending  series, 
and  vice  versa,  therefore  the  general  character  of  a  figure, 
phrase,  or  any  succession  of  tones  is  exaltation  or  relaxation 
in  proportion  as  it  is  on  the  whole  ascending  or  descending. 

11.  The  phrase  may  consist  of  a  single  figure,  or  of  figure 
and  repetition,  or  of  figure  and  treatment,  or  of  different 
figures,  or  of  treatments  of  same  or  different  figures  (a  treat- 
ment being  merely  some  variation  of  a  preceding  figure). 

12.  While  X.\\Q  Jigure  stands  in  the  relation  of  theme  to  the 
pJirase  and  through  the  phrase  to  the  entire  period,  yet  the 
phrase  itself  stands  to  the  period  directly  in  the  relation  of  a 
more  comprehensive  theme. 

13.  The  period  usually  contains  four  phrases;  but  there 
is  no  fixed  mold  in  which  the  period  is  cast,  either  as  to  its 
rhythmic  or  thematic  (especially  thematic)  structure.  It 
should  be  well  proportioned  as  to  unity  and  variety:  unity 
involving  a  principal  idea  binding  the  whole  together;  variety 
involving  thematic  treatment,  and  often  contrasting  themes. 

14.  The  independent  period  (not  dependent  on  the  pre- 
ceding or  following  period)  is  commonly  constructed  as  fol- 
lows:  \.\\Q:  first  phrase  sets  forth  the  principal  theme,  which 
should  excite  anticipation  ;  the  second  phrase  is  of  the  nature 
of  a  partial  response;  the  third  ]}\w:x9.c  is  usually  a  repetition 
of  the  first  phrase  (sometimes  in  a  modified  form);  and  the 
last  phrase  is  a  complete  and  final  response.      (See  Fig.  28.) 

15.  The  ending  or  beginning  of  a  period  is  sometimes  in- 


STRUCTURE    OF    MUSIC  1^7 

fluenced  through  its  connection  with  the  following  or  preced- 
ing period  (by  the  leading  to  or  from). 

1 6.  The  combination  of  periods  into  musical  forms  will  be 
treated  under  "  Form." 

FORM 

1.  Up  to  this  point  we  have  been  dealing  mostly  with  the 
material  out  of  which  musical  compositions  are  constructed.  \ 

2.  Form  deals  with  the  arrangement  of  this  material  into 
definite  shapes  called  musical  forms. 

3.  Just  as  poetic  forms  consist  of  syllables  combined  into 
feet,  feet  into  lines,  lines  into  stanzas,  and  stanzas  into  com- 
plete poems,  so  musical  forms  consist  of  notes  combined  into 
figures,  figures  into  phrases,  phrases  into  periods,  and  periods 
into  complete  movements. 

4.  In  setting  a  poem  to  music,  the  form  of  the  music  must 
be  made  to  fit  the  form  of  the  poem.  The  sentiment  of  the 
music  must  also  correspond  to  the  sentiment  of  the  poem. 

5.  If  the  stanzas  of  the  poem  are  similar  in  sentiment  and 
form,  the  same  music  may  be  repeated  to  each  stanza.  This 
is  called  the  Strophe  Form. 

6.  But  if  the  stanzas  differ  in  sentiment,  the  same  music 
would  not  be  suitable  to  each,  and  therefore  the  music  should 
be  set  to  the  poem  from  beginning  to  end.  This  is  called 
the  Art  Song  Form  (or  through  composed). 

7.  These,  however,  are  only  incidentally  called  forms,  as 
they  do  not  refer  to  the  internal  structure  of  the  music. 

8.  Form  in  its  strict  sense  refers  to  the  structural  shape  of 
a  piece  of  music,  and  therefore  involves  the  internal  structure 
of  the  period,  the  combination  of  periods  into  movements, 
and  movements  into  more  extended  compositions. 

9.  As  the  structure  of  the  period  has  already  been  con- 
sidered, it  only  remains  to  consider  the  combinations  of 
periods  and  of  movements.  A  composition  consists  of  as 
many  movements  as  there   are  actual  changes  in  time,  either 


138  MUSICOLOGY 

as  to  the  time  value  of  the  measure  (see  p.  22  :  6)  or  the  tempo 
(see  p.  30:4). 

10.  All  form  (music  or  other)  is  based  on  the  principles  of 
unity,  contrast,  and  symmetry.  Unity  requires  that  the  parts 
be  so  related  and  connected  as  to  form  one  complete  whole ; 
but  without  contrasts  the  whole  would  be  a  shapeless  mass. 
Contrast  is  necessary  to  remove  too  much  sameness,  thus 
giving  shape  and  character  to  the  whole. 

11.  Unity  and  contrast  tend  to  counteract  each  other,  and 
symmetry  is  their  proper  balance, 

12.  In  musical  forms  unity  is  attained:  first,  by  using  a 
principal  subject,  or  theme,  which  is  so  interwoven  through- 
out as  to  bind  the  parts  together  and  determine  the  general 
character  of  the  whole ;  second,  by  \\  elding  the  parts  to- 
gether— i.e.,  causing  each  part  to  lead  naturally  into  the 
next,  either  directly  or  by  a  connecting  passage. 

13.  Contrast  is  attained:  first,  by  using  secondary  sub- 
jects in  contrast  to  the  principal  subject — but  these  must  be 
less  proniinent  by  being  used  less  frequently,  and  each  occur- 
rence being  followed  by  a  return  to  the  principal  subject ; 
second,  by  development  of  the  subject-matter;  third,  by 
change  of  key  ;  fourtJi,  by  change  of  movement. 

14.  Symmetrical  form  is  the  result  of  a  proper  balance  in 
the  use  of  the  above  means. 

15.  The  smallest  distinct  form  is  the  figure.  Figures  are 
combined  into  larger  forms  called  phrases,  and  phrases  into 
larger  forms  called  sections,  and  sections  into  larger  forms 
called  periods.  But  in  the  period  we  recognize  the  first  com- 
plete and  independent  form. 

16.  The  Song  Form.  The  simplest  complete  musical  form 
is  the  song  form  of  one  period. 

17.  There  are  also  song  forms  of  two  and  three  periods. 
The  spirit  of  the  first  period  (^in  forms  of  more  than  one 
period)  stands  to  the  whole  in  the  relation  of  principal  sub- 
ject.     (By  way  of  distinction,  the  principal   subject   may  be 


STRUCTURE    OF    MUSIC 


139 


called  the  theme,  and  the   secondary  subjects  may   be  called 
episodes.) 

18.    The  Song  Form  may  be  outlined  thus: 
Of  one  period — all  single  church  tunes  or  chorals 


Of  two  periods 


f  1st  period — theme 


All  double  church 
tunes  and  many 
secular  songs;  the 


1  2d  period— episode  and    return  of    second      period 


part  of  theme  as  close 


sometimes  taking 
the  form  of  a 
chorus. 


Of  three  periods 


f  1st  period — theme  1  Most    songs    with 

I  2d  period— episode  in  related  key  I  chorus;      the     3d 
i  3d  period-return     of    theme     in  (Period         usually 
„*^    .    ,  I  taking  the  form  of 

I     Tonic  key  i  a  chorus. 

19.  The  Scherzo,  Minuet,  or  Applied  Song  Form.  This  is 
a  combination  of  song  forms.  The  Schc7-.':o  (meaning  sport, 
jest)  is  a  sprightly,  animated  piece  of  playful  character.  The 
Minuet  is  an  ancient,  slow,  stately  dance.  The  minuet  is  in 
triple  rhythm,  while  the  scherzo  or  applied  song  may  be  in 
any  rhythm.  Their  forms,  however,  are  similar  and  may  be 
outlined  thus : 

I.  Theme  (sometimes  ending  in  related  key). 
Repeat. 

f  Modulatory 


C 

0 

1st  part- 

-0 

a, 
< 

theme 

3 

2d  part — 
trio 

C 

(So    called     be- 

S 

cause   formerly 

0" 

written  in  three 

N 
(L) 

part  harmony) 

2.  Either  an  episode   or 
a  devi'lopweiit  of  thevu 


with 


return   to    theme   in 


Repeat. 


I  Tonic  key 


\  I.    New  theme  (in  related  key  to  theme  o.  ist 
part). 
Repeat. 

1'  Modulatory,       with 
'  close     in    Tonic    of 
[  trio. 
Repeat. 


.    Either  an  episode  or 
a  JevelopmeJit  of  theiiw. 


3   -^ 

3    o 
C    t/1 

^•3 


?d  o. 


3   p 


3d  part— theme  (same  as  ist  part  without  repeats). 

20.  The  above  form  may  be  applied  in  full  to  the  Scherzo; 
but  omitting  the  italic  parts,  it  would  more  properly  be  called 
Minuet  form;  and  omitting  also  the  repeats,  it  would  more 
properly  be  called  Applied  Song  form. 


I40 


MUSICOLOGY 


2  1,  The  minuet  is  also  distinguished  by  its  dance-like 
character.  It  may  be  said  of  the  scherzo  that  it  is  very 
pliable  as  to  form,  and  is  sometimes  written  in  the  rondo  or 
the  sonata  form  ;  so  that  it  is  rather  a  style  than  a  form, 
being  distinguished  by  its  sprightly,  playful  character. 

22.  Observe  that  the  preceding  form  consists  of  two  dis- 
tinct parts  which  stand  related  to  the  whole  as  general  theme 
and  episode,  just  as  in  each  part  the  theme  and  episode  stand 
related  to  the  part;  the  general  theme  binding  the  whole, 
just  as  the  theme  of  each  part  binds  the  part,  giving  the 
sense  of  unity  and  completeness  in  each  case. 

23.  The  Rondo  Form.  The  term  rondo  is  derived  from  a 
l-'rench  poetic  form  in  which  the  first  verse,  after  being  fol- 
lowed by  a  second,  is  repeated. 

24.  The  rondo  is  a  musical  composition  in  which  the  princi- 
pal theme  returns  after  every  digression  or  episode.  There 
are  several  kinds  of  rondos,  which  may  be  outlined  as  follows  : 


1st  rondo  form 


2d  rondo  form 


j  Theme  (usually  a  song  form). 
Episode  (or  passages  in  a  related  key). 
Theme. 


Theme. 

1st  episode  in  any  related  key. 

Theme. 

2d  episode  in  any  other  related  key. 

Theme. 


1 


3-     E; 


;d  rondo  form 


1st  part 


2d  or 


Theme. 

ist   episode,    usually    in     Dom. 
I       key. 
j  Theme  (or  a  conclusion). 

2d  episode  or  a  de-  \ 

velopmcnt  of  isi  '- Modulatory 
middle  part  .  \ 

^  part.  ) 


C  Theme, 
3d  part       I  1st  episode,  now  in  Tonic  key. 
[  Theme  (or  a  conclusion). 

Some  rondos  deviate  more  or  less  from  the  above. 


=.5"  3 


-I    o 

3  3r„- 


3      5- 


!/>  — 


STRUCTURE    OF    MUSIC 


141 


25.  The  Sonata  Form.  This  is  the  climax  of  musical  form. 
A  sonata  is  an  instrumental  composition  consisting  usually  of 
three  or  four  distinct  movements  (differing  in  tempo  or 
rhythm).  The  first  movement  determines  the  character  of 
the  whole,  and  is  therefore  called  the  Sonata  niovonoit. 

26.  The  sonata  form  in  four  movements  may  be  outlined 
thus : 

r  I.  Theme.  .  ,  .        1 

I  n     l^^''    '■^^-    niajor, 

2.  hpisode,     usual!  v  ;  .^  .,  .     . 

I  .  ^  •'  I  II   theme  is  in  a 

]  in  Doni.  key 


'  1st  part 


1st,  or 
sonata 
movement 

{allegro) 


3.   Final  group 


Middle 
part 


Repetition 


2d,    or    slow     movement 
(usually,  grave,  lai-go,  ada- 
gio, or  andante) 
Generally  in  a  related  key 
to  that  of  ist  movement 


[  minor  key. 

consisting  of  one  or 
more    closing    epi- 
sodes, alsoin  Dom. 
key. 
1 1,  2,  and  3  are  now  repeated. 

f  Development  (usually  contrapuntal)  of 
the    theme    and    episode    of    ist    part 

I  (called  the  Free  Fantasia,  because  not 

I  confined  to  any  form),  modulating 
through  several  keys  and  returning  to 

[Tonic  key. 

Theme. 

Episode — now  in  Tonic  key. 
Final  group — now  in  Tonic  key. 
(Sometimes  an  appendix,  or  finale,  is 
\  added.) 

fThis  movement  has  no  fixed  form  but 
may  be  in  any  of  the  following  forms: 
1st — the  sonata  movement  form  (as 
above). 

(  A  series    of   pieces 
the  variation   j  in  which  the  theme 


2d- 

form  j  is      changed 

[  varied. 
3d — the  second  rondo  form. 


and 


3d  movement — a  scherzo  or  a  minuet. 


4th  movement 


("Usually  in  the  rondo  form;  but  some- 
I  times  in  the  sonata  movement  form 
I  or  the  variation  form. 


IJ 

P 

pT 

y 

Hi 

•Jl 

n' 

r/3 

0 

3 

0 

P 

< 

5' 

n 

p 

H 

:■" 

in 

(fl 

^< 

' 

3 

"H- 

\^ 

0 

3 

0 

cr 

Ul 

n 

^ 

n 

3 

0 

3 

c 

n 

i/i 

n 

n' 

^ 

142  MUSICOLOGY 

27.  If  an  entire  sonata  were  in  one  movement  it  would  be- 
come very  wearisome,  but  the  different  movements  are  so 
contrasted  that  each  comes  in  as  a  relief  to  another;  thus  the 
slow  movement  comes  in  like  a  lull  in  a  storm,  and  the  3d 
movement  is  preparatory  to  the  final  climax  in  the  last  move- 
ment. Sometimes  the  scherzo  or  minuet  is  placed  before  the 
slow  movement.  In  three-movement  sonatas  the  scherzo  or 
minuet  is  usually  omitted. 

28.  Suite  Form.  The  old  suite  is  a  succession  of  old  dance 
movements  arranged  with  reference  to  contrast,  and  which 
gradually  assumed  the  following  order: 

(Old  Dances.) 

c  fi.  Allc7naude — a  cheerful  movement  in  even   rh\'thm. 

o  I  2.    Courantc — a  rapid,  running  movement  in  triple  rhj'thm. 
^  •  3.   Sarabande — a  dignified,  stately  movement  in  triple  rhythm. 
.ti     4.   Bott7ree — a  bright,  quick,  hearty  movement  in  even  rhythm. 
CO  [  5.    Gigue — a  very  rapid  movement,  possessing  a  rough  heartiness. 

29.  Other  old  dance  movements  sometimes  used  are: 
chaconne,  a  slow  movement,  generally  major,  and  usually  in 
triple  rhythm ;  passacaglia,  a  rather  bombastic  movement  in 
triple  rhythm,  closely  resembling  the  chaconne,  but  usually 
minor;  viiniict,  a  slow  movement  in  triple  rhythm,  danced 
with  mincing,  dainty  steps,  hence  the  name ;  gavotte,  a  genial 
skipping  movement  in  even  rhythm,  usually  beginning  on 
the  third  beat,  thus  producing  a  mild  syncopation;  pavane, 
a  slow,  stately  movement,  similar  to  the  sarabande,  but  in 
even  rhythm  ;  rigandon,  a  lively  movement  in  even  rhythm, 
beginning  on  the  third  or  fourth  beat,  and  is  sometimes  sung 
as  well  as  danced. 

30.  In  the  modern  suite,  more  modern  movements  (not 
always  dances)  are  frequently  substituted  for  the  old  dance 
movements. 

31.  The  suit e  \% 'OiXQ.  oldest  form  in  which  different  move- 
ments are  combined,  and  is  therefore  the  origin  of  the  sonata 
and  other  modern  forms. 


STRUCTURE    OF    MUSIC  I43 

32.  Contrapuntal  Forms.     (See  pp.  126-129.) 

33.  The  foregoing  are  the  principal  recognized  forms,  and 
with  slight  changes  practically  cover  nearly  all  varieties  of 
music,  and  are  therefore  sufficient  as  a  general  analysis  of 
form. 

34.  The  common  practice  of  calling  nearly  every  slight 
distinction  or  difference  a  form  leads  to  a  great  deal  of  con- 
fusion, and  any  attempt  at  outline  is  vague  and  unsatisfac- 
tory. 

35.  There  are  also  certain  introductory,  intermediate,  and 
concluding  parts  or  passages,  which  arc  not  generally  regarded 
as  an  essential  part  of  the  form  of  a  composition,  as  they 
are  sometimes  used  and  sometimes  omitted.  They  may  be 
classified  thus : 

I  Prelude  1   .  •     i     •    .     j      .• 

I  A    musical    introduction    to    a  more 
Introduction  ■  .  •.■ 

T    .      ,      .  „  important    composition,    or    a     piece 

Introductory  -  Overture  '-         ,  .  . 


used  as  an  opening    to  an    oratorio, 
opera,  or  concert. 


Intrada 
Voluntary,  etc 

finterlude  lA    piece    of    music     placed    between 

Intermediate  I  Intermedium       [more  important  compositions  or  be- 

[  Intermezzo  J  tween  the  acts  of  an  opera. 

f  Postlude  ] 

I  Conclusion  I  The  supplementary  ending  of  a  corn- 
Concluding                 jcoda  ("position. 
I  Finale  J 

36.  The  general  definitions  given  above  sufficiently  de- 
scribe the  general  character  of  each  division  without  going 
into  distinctions,  as  the  aim  here  is  merely  outline. 

37.  The  terms  of  each  division  are  sometimes  used  inter- 
changeably. They  are  not,  as  a  rule,  confined  to  any  fixed 
form.  They  should,  in  general,  be  proportioned  in  length  to 
the  composition  to  which  they  are  attached. 


144 


MUSICOLOGY 


CLASSIFICATION    OF    MUSIC 


I.    Music  may  be  classified  as  follows 
Sacred 


Vocal 

(with  or  without  in- 
strumental accom- 
paniment) 


j  Hymn,  Psalm,  Chant,  Choral,  Anthem, 
/  Antiphony,  Motet,  Mass,  Oratorio,  etc. 


Instrumental 


j  Ballad,    Glee,   Madrigal,  Opera,  Opera 

(  Bouffe,  Grand  Opera,  etc. 
Sacred  or  j  Song,  Recitative,  Aria,  Arietta,  Chorus, 
Secular      I  Cantata,  Cantatilla,  etc, 

(  Sonata,  Sonatina,  Symphony, Concerto. 
■\  Concertino,  Suite,  Overture,  Nocturne, 
(  Fantasia,  Capriccio,  Scherzo,  etc. 


Secular 


Classical 


Dances 


Old 
Dances 


Modern 
Dances 


As  to  number  of  voices    j  Duet,    Trio,    Quartet 
or  instruments  (  Quintet,  Sextet,  etc. 


Allemande,  Courante,  Sara- 
bande,  Bourree,  Gigue,  Cha- 
conne,  Passacaglia,  Minuet, 
Gavotte,  Pavane,  Rigaudon, 
etc. 

r  Polonaise,  Mazurka,  Polka, 
I  Schottische,  Waltz,  March, 
I  Quickstep,  Galop,  Fandango, 
L  Reel,  Quadrille,   Cotillon, etc. 

I  Called.Chamber  Music 
I  when  written  for 
■\  stringed  instruments, 
j  or  for  piano  and  other 
[  instruments. 


2.  Sacred  Vocal  Music.  The  Hyjnn  is  a  song  of  praise  or 
thanksgiving  to  God.  A  Psa/in  is  a  metrical  translation  of 
one  of  the  Psalms  set  to  music.  A  Chant  consists  of  words 
recited  to  musical  tones  without  musical  measure.  The 
Clioral  is  a  psalm  or  hymn  sung  in  unison  by  the  congregation. 
An  Antheni  is  a  setting  of  scriptural  texts  to  music.  The 
Antiphony  is  the  most  ancient  form  of  church  music,  and  con- 
sists of  responsive  singing,  one  part  or  choir  answering  or  re- 
sponding to  another.  The  motet  is  also  a  species  of  hymn 
with  a  scrij)tural  text  or  texts,  designed  for  chorus  or  for 
choruses  interspersed  with  solos;  it  is  usually  livelier  and  more 
brisk   than   other   religious   music    and   is   sometimes  used  as 


STRUCTURE    OF    MUSIC  1 45 

synonymous  with  anthem.  The  Mass  is  a  musical  composi- 
tion used  in  the  CathoHc  Church  in  celebrating  mass,  hence 
the  name.  The  Oratorio  is  a  sacred  musical  drama,  and  con- 
sists of  recitatives,  choruses,  solos,  duets,  trios,  etc.,  accom- 
panied by  orchestra. 

3.  Secular  Vocal  Music.  A  Ballad  is  a  narrative  told  in 
song.  The  Glee  is  a  composition  of  a  light  character,  usually 
sung  by  one  voice  to  a  part,  and  generally  unaccompanied. 
The  Madrigal  is  a  composition  for  voices  without  accompani- 
ment, each  part  being  supported  by  several  voices.  The 
Opera  is  a  secular  musical  drama,  and  consists  of  an  overture, 
arias,  choruses,  recitatives,  duets,  trios/  etc.,  accompanied  by 
scenery  and  dramatic  action.  If  the  character  is  serious  it  is 
called  a  Grand  Opera.  If  the  character  is  comic  it  is  called 
an  Opera  Douffe,  or  comic  opera.  The  Operetta  is  a  short, 
light  opera. 

4.  Sacred  or  Secular  Vocal  Music.  Song  is  a  general 
term  applicable  to  all  vocal  music,  whether  of  human  beings 
or  of  birds,  but  most  usually  suggests  a  simple  melody  re- 
peated to  each  stanza  of  a  sacred  or  secular  poem.  The  Reei- 
tative  is  a  musical  declamation  imitating  declamatory  speech 
with  the  singing  voice,  usually  with  instrumental  accompani- 
ment. The  Aria,  in  its  general  sense,  is  an  air,  or  melody  ; 
more  strictly,  it  is  an  accompanied  solo,  usually  similar  in 
form  to  a  minuet.  The  Arietta  is  a  small  aria.  The  Chorjis 
is  a  composition  for  numerous  voices.  The  Cantata  is  a 
composition  mixed  with  recitatives,  arias,  and  choruses,  with 
instrumental  accompaniment.  The  Cantatilla  is  a  small  can- 
tata. 

5.  Classical  Instrumental  Music.  The  Sonata  (see  p. 
141  :  2$).  The  Sonatina  is  a  small  sonata.  The  Symphony 
is  a  sonata  for  full  orchestral  accompaniment.  The  Concerto 
is  usually  a  sonata  in  three  parts,  in  which  one  or  several 
concerting  instruments  play  the  principal  parts  accompanied 
by  an  orchestra.      The   Coneertino  is  a  small   concerto.      The 


146  MUSICOLOGY 

Suite  (see  p.  142  :  28).  The  Overture  is  a  composition  used 
as  an  opening  to  an  oratorio,  opera,  concert,  or  drama,  and  is 
usually  in  the  form  of  the  Sonata  Movement.  The  Concert 
Overture  is  an  independent  composition  for  concert  perform- 
ance. The  Nocturne  is  a  composition  of  the  character  of  a 
calm  night,  hence  its  name.  The  Fantasia  is  a  composition 
for  solo  instrument  written  without  regard  to  the  restrictions 
of  form.  The  Capriccio  is  a  composition  written  in  a  capri- 
cious or  free  st-'le — a  species  of  fantasia.  The  Scherzo  (see  p. 
139:  19V 

6.  Old  Dances.      (See  p.  142  :  28,  29.) 

7.  Modern  Dances.  The  Polonaise  and  the  iMacurka  are 
Polish  national  dances.  They  are  written  in  4  time.  The 
Polonaise  is  slow  and  stately  and  is  more  of  a  walking  or  step- 
ping than  a  dance.  The  Mazurka  is  lively  and  of  a  senti- 
mental character.  The  Polka  is  a  skipping  \  movement  of 
Bohemian  origin.  The  Schottische  is  similar  to  the  polka,  but 
somewhat  slower.  The  Waltz  is  a  circular  whirling  dance  in 
4  time.  The  March  is  a  movement  suited  to  guide  the  walk- 
ing of  masses.  The  Quickstep  is  a  quick,  lively  march.  The 
Galop  is  a  quick  dance  tune.  The  Fandango  is  a  lively  Spanish 
dance  tune  in  g  or  §  time.  The  Reel  is  a  lively  Scottish  dance 
tune.  The  Quadrille  is  a  French  dance  tune  performed  by 
four  couples  placed  in  quadrangular  position,  hence  the 
name.  The  Cotillon  is  a  lively,  animated  dance  tune,  usually 
in  %  time. 

8.  Every  dance  has  its  own  particular  rhythm,  which  gives 
it  its  character. 


PART    THIRD 


ACOUSTICS 


1.  Acoustics  treats  of  the  laws  of  sound.  Sound  is  caused 
by  vibrations.  If  the  vibrations  are  regular  a  musical  tone  is 
produced,  while  noise  is  the  result  of  irregular  vibrations. 

2.  The  range  of  sound  is  from  about  i6  vibrations  per 
second  to  about  38,000  per  second  (about  eleven  octaves),  as 
these  limits  represent  the  capacity  of  the  ear  to  respond. 
The  range  of  the  seven-octave  piano  is  from  about  32  to  4100 
vibrations  per  second. 

3.  The  velocity  with  which  sound  travels  may  be  deter- 
mined by  the  simple  method  of  observing  the  time  between 
seeing  a  gun  fired  and  hearing  the  report  (the  time  for  light  to 
travel  being  inappreciable) ;  then  by  calculations  based  on  the 
distance  and  time  we  may  determine  the  velocity,  which  is 
thus  found  to  be  about  1 100  feet  per  second,  being  influenced 
somewhat  by  the  state  of  the  atmosphere.  It  has  also  been 
found  by  tests  that  all  sounds  travel  with  the  same  velocity. 

4.  Sound  travels  through  the  air  in  waves  similar  to  waves 
of  water,  except  that  the  waves  are  to  and  fro  instead  of  up 
and  down.  As  the  pressure  of  the  air  is  in  all  directions  there 
is  only  room  for  compression  and  expansion,  thus  producing 
to-and-fro   waves,    which   circle    outward    in    all     directions. 


148  MUSICOLOGY 

The  vibrations  of  the  sounding  body  are  transmitted  to  the 
air,  producing  similar  vibrations  in  the  particles  of  air,  each 
impulse  passing  onward  from  one  particle  to  the  next.  Each 
particle  of  air  passes  through  every  possible  phase  (position) 
of  vibration  during  one  complete  vibration. 

5.  The  length  of  a  sound-wave  is  the  distance  between  any 
two  particles  in  exactly  the  same  phase,  including  between 
them  every  possible  phase  of  one  complete  vibration.  The 
waves  thus  advance,  while  the  particles  of  air  merely  vibrate. 

6.  Since  the  velocity  of  sound  is  about  1 100  feet  per 
second,  if  we  divide  1 100  by  the  number  of  vibrations  cor- 
responding to  any  pitch  we  would  get  the  wave-length  of  that 
pitch.  We  would  thus  find  that  the  wave-lengths  correspond- 
ing to  all  the  tones  of  the  seven-octave  piano  would  range 
from  about  three  inches  to  about  thirty-four  feet. 

7.  The  Laws  of  Vibration  are  most  easily  studied  by  means 
of  taut  strings. 

8.  If  a  taut  string  is  plucked  (as  in  the  guitar),  bowed  (as 
in  the  violin),  or  struck  (as  in  the  piano),  it  will  be  seen  to 
vibrate  from  side  to  side ;  the  tone  produced  depending  upon 
the  length,  tension,  thickness,  and  weight  of  the  string. 
Three  things  will  be  noticed  :  loudness,  pitch,  and  cjuality. 
The  loudness  depends  on  the  force  of  the  stroke  (or  the 
amplitude  of  the  vibrations) ;  \\\e pitcJi  depends  on  the  number 
of  vibrations  per  second ;  and  the  quality  depends  on  the 
compound  nature  of  the  vibrations  (all  strings  vibrate,  not 
only  as  a  whole,  but  also  in  parts  which  produce  overtones 
upon  which  depends  the  quality  of  tone,  as  all  simple  tones  of 
the  same  pitch  sound  alike). 

9.  A  Sonometer  is  an  instrument  for  studying  the  vibrations 
of  strings.  It  consists  of  a  long  resonance  box  over  which  one 
or  more  strings  may  be  stretched  by  means  of  weights  and 
pulleys,  and  provided  with  a  graduated  scale  and  a  movable 
bridge  so  that  the  exact  vibrating  length  of  the  string  may  be 
measured. 


ACOUSTICS  149 

10.  By  means  of  the  sonometer  the  following  laws  have 
been  demonstrated  : 

1 1.  First  Law.  The  shorter  the  string  the  faster  it  vibrates 
in  inverse  proportion  to  the  length  (shortening  the  string 
to  I,  \,  \,  \,  etc.,  increases  the  number  of  vibrations  2,  3,  4, 
5,  etc.,  times). 

12.  Second  Law.  The  tighter  a  string  is  drawn  the  faster 
it  vibrates,  but  in  direct  proportion  to  the  square  root  of  the 
tension  (thus,  increasing  the  tension  4  times,  9  times,  etc.,  in- 
creases the  number  of  vibrations  2  times,  3  times,  etc.). 

1 3 .  Third  Law.  TJie  thicker  the  string  the  slozver  it  vibra  tes  in 
inverse  proportion  to  the  thickness  (thus,  doubling  the  diameter 
of  the  string  diminishes  the  number  of  vibrations  one-half). 

14.  Fourth  Law.  The  greater  the  density,  or  weight,  of  the 
string  the  slower  it  vibrates  in  inverse  proportion  to  the  square 
root  of  the  weight  (thus,  multiplying  the  weight  by  four  divides 
the  number  of  vibrations  by  two).  Strings  are  sometimes 
coiled  with  wire  to  increase  their  weight. 

15.  Fifth  Law.  The  intensity,  or  loudness,  is  in  direct  pro- 
portion to  the  square  of  the  amplitude  {extent)  of  vibration 
(thus,  doubling  the  amplitude  increases  the  loudness  four 
times).  The  time  of  vibration  is  not  affected  by  the  ampli- 
tude of  vibration,  provided  other  conditions  remain  the  same: 
just  as  the  time  of  each  swing  of  a  pendulum  is  not  affected 
by  the  amplitude,  or  extent,  of  the  swing,  provided  the  length 
of  the  pendulum  remains  the  same. 

16.  Compound  Tones.  If  a  taut  string  be  struck,  its  fun- 
damental   tone    is    due    to    its  vibrations   as  a  whole,  thus: 


tone.      Now  if  the 


which  we    will  suppose  is  this,  f^^ 

string  be  lightly  touched  at  its  middle  point  (just  enough  to 


i;o 


-MUSICULUGV 


prevent  its  fundamental  vibrations  but  not  to  push  it  aside 
from  its  straight  position)  and  made  to  vibrate,  it  will  vibrate 
in   halves,  thus: 


(called  segments  or  loops,  and  the  dividing  point  a  node); 
each  part,  being  half  the  length  of  the  string,  will  vibrate  twice 
as  fast  (Law  i);    the  tone  produced  will  be    the  octave  of  the 

fundamental   tone,  which  is    this,    -^- 


A.      Again,  if   the 


string  be  touched  at  one-third  its  length  and  the  shorter  part 
struck,  the  whole  string  will  vibrate  in  thirds,  thus: 

each  part  being  one-third  will  vibrate  three  times  as  fast  as 
the  whole  string,  and  the  tone  produced  will  be  a  perfect  5th 


above  the   first   octave,  which   is  this,  -(^  Again,    if 

the  string  is  touched  at  one-quarter  its  length  and  the  shorter 
part  struck,  it  will  vibrate  in  fourths,  thus: 

and  each  part,  being  one-fourth  the  length  of  the  string,  will 
vibrate  four  times  as  fast,  and  will  sound   the  second   octave 


of  the  fundamental,  which  is  this,    -^' 


Again,  if  the 


string  be  touched  at  one-fifth  its  length  and  the  shorter  part 
struck,  it  will  vibrate  in  fifths,  thus: 

each  part  vibrating  five  times  as  fast  as  the  fundamental,  and 
sounding  the  major  third   above  the  second   octave,  which  is 


this,  rac 4      We  may  continue  thus  to   divide  the  strine 


indefinitely. 


I 


ACOUSTICS 


151 


17.  Now  if  the  string  be  made  to  vibrate  free,  it  will  not 
only  vibrate  as  a  whole  but  will  at  the  same  time  vibrate, 
more  or  less  (varying  somewhat,  as  to  relative  proportions, 
with  character  of  string  and  place,  manner,  and  force  of  its 
excitation),  in  halves,  thirds,  fourths,  etc.,  indefinitely. 

18.  By  trying  to  conceive  of  the  string  vibrating  as  a,  b,  c, 
d,  and  c  at  the  same  time  (the  whole  like  a  large  wave  bear- 
ing on  its  surface  smaller  waves,  and  these  still  smaller,  etc.), 
we  may  form  an  approximate  idea  of  the  compound  nature 
of  the  vibrations  and,  in  turn,  of  the  compound  nature  of  the 
resulting  tone,  which  is  a  compound  of  the  fundamental  tone 
and  the  overtones  produced  by  the  vibrating  segments — the 
overtones  growing  weaker  as  they  ascend  in  pitch.  Some  of 
the  lower  overtone  vibrations  may  be  plainly  seen  on  the 
lowest  strings  of  almost  any  stringed  instrument,  but  the 
higher  overtone  vibrations  are  too  rapid  to  be  seen. 

19.  Any  fundamental  tone 
with  its  overtones  is  called  a 
Harmonic  Series.  Fig.  29 
shows  the  harmonic  series  of 
great  C.  The  series  is  the 
same  for  all  tones,  differing 
only  in  pitch. 

20.  Any  harmonic  series  is 
called    a    chord    of  nature,    as 
the  compound  tone  thus  pro- 
duced   may    be  considered  in  Pm.  39. 
itself  a  natural  chord  or  combination  of  tones. 

21.  By  the  pitch  of  a  compound  tone  is  meant  the  pitch  of 
its  fundamental,  this  being  the  absorbing  or  ruling  tone. 

22.  The  overtones  produced  by  the  uneven  divisions  of  the 
string  (thirds,  fifths,  etc.)  are  easier  to  hear  than  those  pro- 
duced by  the  even  divisions  (halves  and  fourths,  forming 
octaves,  so  completely  blend  into  the  fundamental  as  to  be 
practically  absorbed).      The  perfect  5th  above  the  first  octave 


s 


152  MUSTCOI.OGY 

(produced  by  the  string  vibrating  in  thirds)  is  the  easiest 
heard,  and  after  this  the  major  3d  above  the  second  octave 
(produced  by  the  string  vibrating  in  fifths).  Lightly  sound- 
ing the  note  we  wish  to  distinguish,  previously  to  sounding 
the  fundamental,  will  assist  in  hearing  it  in  the  combination. 

23.  It  is  not  to  be  inferred  that  the  overtones  are  weak 
because  they  are  difficult  to  hear;  the  difficulty  is  due  to  the 
blending.  The  lower  overtones  are  usually  quite  prominent 
in  the  best  qualities  of  tone.  They  increase  in  prominence 
descending  the  scale.  At  below  40  vibrations  per  second 
the  pitch  is  quite  indefinite  owing  to  the  prominence  of  the 
lower  overtones. 

24.  Siviple  tones  are  those  free  from  overtones,  and  due, 
therefore,  to  simple  vibrations.  Simple  vibrations  are  those 
without  segmental  divisions,  and  are  also  called  pendular 
vibrations  because  similar  in  character  to  a  swinging  pendulum. 

25.  It  is  nearly  impossible  to  produce  tones  free  of  over- 
tones (simple  tones).  The  nearest  approach  to  simple  tones 
is  the  tone  produced  by  a  tuning-fork,  the  lowest  tones  of  the 
larger  stopped  pipes  of  the  pipe-organ,  and  the  lowest  tones 
of  a  wooden  flute. 

26.  The  practical  effect  of  overtones  is  to  give  quality  or 
tone-color  to  sounds.  A  chord  of  nature  (harmonic  series)  is 
a  compound  of  its  different  tones,  just  as  light  is  a  compound 
of  the  seven  colors  of  the  rainbow,  which  blending  together 
in  their  natural  proportion  produce  white  light,  but  any 
deviation  from  this  natural  proportion  produces  a  shade  of 
color  which  partakes  of  the  character  of  the  predominating 
color  or  colors ;  so  the  quality,  or  tone-color,  of  tones  de- 
pends upon  the  proportionate  strength  of  its  various  over- 
tones. Decrease  in  overtones  tends  toward  a  dull,  hollow, 
monotonous  quality,  while  increase  tends  toward  a  bright, 
sharp,  penetrating  quality.  The  lower  overtones  tend  toward 
a  deep,  mellow  quality,  and  the  higher  overtones  toward  a 
thin,  metallic  quality. 


ACOUSTICS  153 

27.  The  overtones  are  influenced  largely  by  the  character 
of  the  string.  The  harmonic  series  will  extend  as  high  as  the 
thickness  and  stiffness  of  the  string  will  permit.  Thin  flexi- 
ble strings  will  naturally  vibrate  in  shorter  lengths  than 
thicker  and  stiffer  ones,  and  thus  produce  higher  overtones. 
The  more  elastic  the  string  the  stronger  all  the  overtones 
will  be. 

28.  The  overtones  are  also  affected  by  the  manner  in 
which  the  string  is  set  in  motion  (whether  struck,  plucked, 
or  bowed) ;  also,  soft  broad  hammers  and  picks  tend  to  soften 
by  preventing  many  of  the  smaller  loops  from  forming,  while 
hard  sharp  hammers  or  picks  have  the  opposite  effect — hence 
the  felted  hammers  of  the  piano,  which  are  softer  and  heavier 
toward  the  lower  tones  and  harder  and  lighter  toward  the 
higher  tones.  The  force  of  the  blow  naturally  affects  the 
extent  of  the  harmonic  series  as  well  as  the  strength  of 
all. 

29.  The  place  where  the  string  is  struck  (or  otherwise  ex- 
cited) is  also  very  important.  No  node  can  form  at  or  very 
near  the  point  struck,  hence  all  overtones  having  a  node  at  or 
very  near  that  point  will  be  missing ;  thus,  if  we  strike  the 
string  at  its  middle  point,  all  the  even  numbered  harmonics 
will  be  missing,  producing  a  hollow,  nasal  twang.  If  we 
strike  the  string  at  one-third  its  length,  every  third  harmonic 
will  be  missing.  If  we  strike  the  string  at  one-fourth  its 
length,  every  fourth  harmonic  will  be  missing,  and  so  on  ; 
and  of  the  harmonics  which  remain,  those  are  naturally  the 
strongest  which  have  the  striking  point  midway  between  their 
nodes,  the  others  being  more  or  less  modified.  On  the  other 
hand,  if  we  touch  a  vibrating  string  at  any  point,  thus  form- 
ing a  node  instead  of  a  loop,  we  damp  all  the  harmonics 
which  have  no  node  at  that  point,  while  those  having  a  node 
at  that  point  will  remain. 

30.  Observe,  from  Fig.  29,  that  the  harmonic  series  from  the 
7th  up  is  inharmonic,    as  the  overtones    by  their  closeness 


154  MUSICOLOGY 

begin  at  this  point  to  form  dissonant  intervals  with  those 
nearest.  However,  the  inharmonic  part  of  the  series  is 
largely,  though  not  wholly,  drowned  in  the  more  prominent 
harmonic  lower  part ;  so  that  the  harmonic  series  is  in  the 
main  harmonic. 

31.  In  pianos  the  strings  are  struck  by  the  hammers  at 
between  one-seventh  and  one-ninth  of  their  length,  thus  de- 
stroying the  7th,  8th,  and  9th  harmonics  (6th,  7th,  and  8th 
overtones).  The  higher  overtones,  being  weak  and  having 
nodes  within  the  influence  of  the  stroke,  are  practically  de- 
stroyed also. 

32.  By  thus  removing  the  inharmonic  part,  the  harmonic 
series  is  made  more  perfectly  harmonic  as  well  as  more  clearly 

^  defined    (being    within     definite 

limits),    as    shown    in    Fig.    30. 

'  This  is  all  of  the  harmonic  series 

, (chord     of    nature)     that    is     of 

— •—  practical   use  or    application   to 


IJ 


f^S\,                  '  music.      (Of    course    this    same 

\J -   series  of  intervals  belongs  to  any 


~  fundamental  tone,  differing  only 


in  pitch.) 

-©-  33.    In    Fig.    30    we    observe 

Fig.  30.  that  wc  have  three  C's,  two  G's, 

and  one  E,  and  that  we  have  the  major  triad — C,  E,  G — di- 
rect on  the  second  octave  of  the  fundamental.  Also,  observ- 
ing the  successive  intervals  from  the  fundamental  up,  we  have 
the  octave,  perfect  5th,  perfect  4th,  major  3d,  and  minor  3d, 
by  which  we  observe  the  relative  prominence  of  these  inter- 
vals in  the  music  scale. 

34.  We  also  notice  that  the  second  octave  is  subdivided 
into  a  perfect  5th  plus  a  perfect  4th;  and  that  the  perfect 
5th  above  is  similarly  subdivided  into  a  major  3d  plus  a 
minor  3d.  The  entire  harmonic  series  being  thus  a  series  of 
diminishing  similar  subdixisions  indcfinitclv  extended. 


ACOUSTICS 


155 


35.  Observe  that  any  harmonic  series  limited  to  the  fifth 
overtone,  as  in  Fig,  30,  consists  of  three  distinct  notes  (or 
letters),  which,  when  sounded  together,  form  a  major  triad  ; 
and  the  root  of  the  triad  is  naturally  the  tone  from  which  the 
triad  is  derived.  The  major  triad  is  the  only  chord  that  has 
thus  a  root  in  the  sense  that  a  plant  has  a  root.  In  the  other 
triads  the  word  "  root  "  is  used  more  in  the  sense  of  ground 
note,  or  lowest  note,  upon  which  the  triad  is  built  in  3ds. 

36.  Major  triads  are  also  called  harmonic  triads,  because 
thus  in  perfect 'harmony  with  the  harmonic  series  of  their 
fundamental  root  notes ;  to  which  fact  is  evidently  due  their 
greater  relative  importance.  The  fundamental  note  of  the 
series  from  which  a  chord  is  derived  is  called  the  fundamental 
bass  of  the  chord  in  distinction  from  the  common  bass,  or 
lowest  written  note. 

37.  It  is  most  convenient  and  usual  to  regard  a  minor  triad 
as  a  major  triad  with  an  intruding  3d  ;  but  by  taking  each 
note  in  turn  as  the  intruding  note,  the  minor  triad  may  be 
regarded  as  derived  from  any  one  of  three  different  harmonic 
series. 

38.  Taking  the  minor  triad,  E,  G,  B,  Fig.  31,  for  ex- 
ample, we  see  that  the  two  lower  notes  belong  to  the  har- 
monic series  on  C,  the  outside 
notes  belong  to  the  series  on  E, 
and  the  two  upper  notes  belong 
to  the  series  on  G ;  so  that  in 
this  sense  the  minor  triad  has 
three  distinct  but  imperfect 
roots,  or  fundamental  basses  (re- 
garding  each   note  of  the  triad 

in  turn  as  the  intruding  note).  LO- 

We  see  that  the  minor  triad  has  Q-&- 

no   perfect   root   with  which   all  Fk,.  31. 

three  of  its  notes  harmonize.      We  notice  that  these  three 

imperfect   roots  together  form  the  major  triad,  C,  E,   G;    the 


n 

/        in 

/ 

Da 

f 

J 

C» 

#- 

r  m 

t) 

-• — 

• 

• 

^^v 

• 

• 

• 

V 

• 

-G-^- 

156 


MUSICOLOGY 


Dim.  Triad. 


Aug.  Triad. 


middle  root  E,  being  the  .same  letter  as  the  lowest  note  of  the 
minor  triad,  is  the  usually  accepted  root,  as  it  conforms  to 
the  simple  uniform  method  of  treating  all  chords  as  built  in 
3ds  on  the  lowest  note  as  root ;  but  all  three  roots  may  be 
taken  interchangeably  as  the  fundamental  bass  of  the  triad. 
All  minor  triads  being  built  alike  may  be  analyzed  with  sim- 
ilar results.  We  see  that  every  minor  triad  is  directly  re- 
lated to  the  major  triad  formed  by  its  three  roots. 

39.  In  Fig.  32  the  diminished  and  augmented  triads  are 
analyzed.      In  either  case,  the  outside  note^  not  forming  a 

perfect  5th,  can- 
not both  belong 
to  the  same  har- 
monic series,  so 
that  these  triads 
have  two  imper- 
fect roots,  or 
f  u  n  d  a  me  n  t  a  1 
basses,  instead  of 
three  (regarding 
the  upper  and 
lower  notes  in 
turn  as  the  in- 
truding note). 

40.  Observe  that  the  roots  of  the  diminished  triad,  B,  D, 
F,  are  G  and  B  b,  neither  of  which  belongs  to  the  triad ;  for 
which  reason  it  is  frequently  treated  as  a  dominant  7th  chord 
with  root  omitted. 

41.  The  roots  of  the  augmented  triad,  C,  E,  Gj|,  are  C  and 
E,  both  of  which  belong  to  the  triad ;  but  C  being  also  the 
lowest  note  of  the  triad  is  the   usually  accepted   root. 

42.  The  7th  chords  (see  outline  on  p.  69)  mostly  receive 
their  justification  from  the  triads  on  which  they  are  based 
(see  exception,  chord  of  added  6th,  p.  yi))' 

43.  The  harmonic  7th  chord  consists  of  a  major  triad  with 


5 

/ 

VII  •• 

Ill' 

(( 

^ 

VL 

Jf. 

B»         •        b» 

C-     —       T 

^a> 

• 

l^    >              .          ' 

• 

^-^                        > 

• 

J* 

-E-^- 


bOBly 


G^ 


GG 


Kig.  33. 


ACOUSTICS 


157 


the  7th  harmonic  added  ;  but  the  7th  harmonic  is  not  ex- 
actly represented  by  any  tone  of  the  scale  in  common  use 
(see  Fig.  29,  p.  151);  its  nearest  representative  is  the  minor 
7th.  The  Vj  chord  being  the  only  chord  with  major  triad 
and  minor  7th  is  therefore  the  nearest  representative  of  the 
harmonic  7th  chord,  and  may  be  regarded  as  a  slightly  de- 
fective harmonic  7th  chord,  hence  the  prominence  of  the  V^ 
chord. 

44.  The  VIIt  and  Vlli  chords  may  be  regarded  as  Vg 
chords  with  root  omitted  (see  p.  72  :  22),  for  the  same  reason 
that  the  vil°  triad  (Fig.  32)  may  be  regarded  as  a  V^  chord 
with  root  omitted. 

45.  We  see  that  the  harmonic  series  is  the  foundation  of 
harmony,  or  chord  formation. 

THE   SCALE   OF  NATURE 

1.  In  Fig.  33  we  have  the  harmonic  series  (chord  of  na- 
ture) on  C,  including  the  fourth  C  ;  and  on  the  G  thus  obtained 
we  have  based  another  series. 
The  figures  of  the  first  series 
show  the  divisions  of  the  string 
and  therefore  the  comparative 
rate  of  vibration  of  the  different 
tones  of  that  series.  The  fig- 
ures of  the  second  series  are 
practically  a  continuation  of  the 
first,  being  based  on  G,  which 
has  three  times  the  vibrations  of 
the  first  base ;  therefore  each 
interval  of  this  series  will  have 
three  times  the  vibrations  of 
the  corresponding  interval  of  the  first  series  (the  series  of  in- 
tervals in  both  being  the  same,  differing  only  in  pitch). 

2.  Between  i  and  2  we  have  the  interval  of  an  octave,  the 
ratio  of  which  (i  :  2)  is  2.     Between  2  and  3  we  have  a  per- 


•18 
*15 
•12 


-•-9- 


-•-6- 


•4- 


s 


•  3 


<g3 


•2 


Fig.  3:3. 


158  MUSICOLOGV 

feet  5th,  the  ratio  of  which  (2  :  3)  is  |.  Between  3  and  4  we 
have  a  perfect  4th,  the  ratio  of  which  (3  :  4)  is  a.  Between  4 
and  5  we  have  a  major  3d,  the  ratio  of  which  (4:  5)  is  |,  Be- 
tween 5  and  6  Ave  have  a  minor  3d,  the  ratio  of  which  (5  :  6) 
is  |.  Between  3  and  5  we  have  a  major  6th,  the  ratio  of 
which  (3  :  5)  is  |.  Between  5  and  8  we  have  a  minor  6th,  the 
ratio  of  which  (5  :  8)  is  |.  Between  8  and  9  we  haxe  a  major 
2d,  the  ratio  of  which  (8 :  9)  is  |.  Between  8  and  15  we  have 
a  major  7th,  the  ratio  of  which  (8:  15)  is  y.  In  the  same 
way  we  find  the  ratio  between  any  two  tones  by  dividing  the 
number  of  vibrations  in  the  higher  tone  by  the  number  in  the 
lower  tone,  and  if  necessary  reducing  the  resulting  fraction  to 
its  lowest  terms. 

3.  We  have  here  the  ratios  of  all  the  natural  intervals  of 
both  the  major  and  minor  diatonic  scales  by  simply  arranging 
them  in  the  order  of  the  interval  numbers,  as  follows: 

n      |n    fn    ^n   #n    ^n    ^/n    2n 
Major  Diatonic  scale— C      DEFGABC!? 

n     |n     |n    |n    |n     |n    ^n    2n 
Minor  Diatonic  scale — C     D      El^     F     G     A''      B      C 

(Beginning  with  A  and  sharping  G  will  also  give  the  minor  scale.) 

Fig.  34. 

4.  Letting  ;/  represent  the  number  of  vibrations  in  C,  we 
may  obtain  the  number  of  vibrations  of  any  degree  of  the 
scale  by  multiplying  n  by  the  corresponding  ratio.  A  scale 
based  thus  upon  the  laws  of  natural  proportions  (as  derived 
from  vibrating  strings)  is  called  a  scale  of  natiire. 

5.  We  may  assume  a  long  wire  with  sufficient  tension  to 
allow  it  to  vibrate  once  per  second.  Though  no  sound  is  pro- 
duced, we  may  call  this  imaginary  note  C;  one-half  the 
length  of  the  wire  will  give  two  vibrations  per  second,  one- 
fourth  will  give  four  vibrations,  one-eighth  will  give  eight 
vibrations,  one-sixteenth  will  give  sixteen  vibrations  per 
second  (the  lowest   sound).     At  the  sixth  subdivision  (sixth 


ACOUSTICS 


159 


octave  from  the  starting  C)  we  get  64  vibrations  per  second, 
corresponding  to  the  lowest  degree  given  in   Fig.  35. 

6.  Now,  multiplying  64  by  the  different  interval  ratios 
already  found,  and  shown  on  the  left  in  the  figure,  we  get  the 
number  of  vibrations  in  each  degree  of  the  first  octave.  Each 
octave  may  be  built  in  the  same  manner  or  by  simply  doub- 
ling the  numbers  in  the  octave  below  successively. 

7.  In  forming  the  minor  scale  we  would  only  have  to  sub- 
stitute the  ratios  of  the  minor 
3d  and  6th  (|  and  |)   instead 
of    the  major    3d    and    6th  (| 
and  I). 

8.  The  standard  of  pitch 
given  in  Fig.  35  (middle  C  256) 
is  cA\e.d  philosophical  ov  physi- 
cal pitch,  because  it  is  based 
on  a  simple  physical  process, 
as  shown  above.  It  is  also 
called  classical  pitch,  because 
it  is  the  average  standard  used  by  the  classical  composers. 
Concert  pitch  is  a  higher  standard  of  pitch.  The  standard 
also  varies  in  different  countries. 

9.  The  tendency  of  manufacturers  of  musical  instruments 
has  been  to  raise  the  standard  of  pitch,  to  secure  greater  brill- 
iancy ;  so  that  middle  C  varies  in  different  standards  from 
256  to  about  270  vibrations  per  second. 

10.  In  Fig.  34,  n  represents  the  number  of  vibrations  in  C. 
We  may  let  «  =  i,  as  in  Fig.  36.  The  relationships  involved 
are  thus  made  to  stand  out  alone. 

Vibration  values —     i       f       t      I       I       I      V      2 
Diatonic  scale—         CDEFGABC 
Consecutive  ratios— i  X  |  X  VXyl  X  I  X  ^/X  |  X  il  =  2 
I''i<i.  :!0. 

11.  The  fractions  above  the  scale  letters  are  the  ratios  of 
the  intervals   from  C   (as   derived   from   the   harmonic   series, 


Fig.  35. 


l60  MUSICOLOGV 

Fig.  33),  and  represent  the  \abration  value  of  each  letter  as 
compared  with  that  of  C.  The  fractions  below  the  letters 
are  the  ratios  of  the  intervals  between  consecutive  letters : 
thus,  the  ratio  D  :  E  is  |  :  f  =  V  I  or,  E  :  F  is  |  :  |  =  || ; 
etc. 

12.  Observe  that  the  lower  fractions  multiplied  together 
are  equal  to  the  octave,  and  that  the  vibration  value  of  each 
interval  from  C  is  equal  to  the  product  of  all  the  ratios  up  to 
that  point.  Descending  the  scale  is  the  reverse  of  ascending; 
therefore  inverting  the  ratios  will  give  the  successive  divisors 
from  right  to  left. 

13.  Owing  to  the  constantly  increasing  rate  of  vibration 
ascending  the  scale,  it  is  evident  that  the  same  interval  will 
vary  as  to  number  of  vibrations  with  pitch,  so  that  intervals 
cannot  be  represented  (except  locally)  by  vibration  differences. 
The  simplest  method,  therefore,  of  expressing  intervals  is  by 
ratios,  since  the  ratio  of  any  interval  does  not  vary  with  pitch. 
Ratios  are  thus  general  in  character,  enabling  us  to  express 
intervals,  or  pitch  differences,  in  a  general  sense. 

14.  Besides,  it  may  be  shown  that  multiplying  by  ratios 
gives  the  same  result  as  adding  differences.  Taking  any  two 
numbers  for  example,  as  3  and  4,  thus:  4  —  3=  i  =  differ- 
ence; 1=1^  =  ratio.  Now,  multiplying  3  by  the  ratio  i^  is 
merely  adding  1^  of  itself ,  or  i,  which  is  the  difTerence.  There- 
fore the  common  conception  of  the  music  scale  as  a  succession 
of  intervals  one  above  the  other  (the  whole  being  naturally 
the  sum  of  all  its  parts)  is  in  no  sense  affected.  Hence,  to 
add  intervals  we  multiply  their  ratios ;  and  conversely,  we 
subtract  by  dividing. 

15.  Observe  in  the  lower  ratios  in  Fig.  36  that  there  are 
only  three  different  kinds  of  ratios  used :  f,  called  major  tone, 
V,  called  minor  tone,  and  -|-|,  called  major  semitone.  Observe 
also  that  the  octave  consists  of  three  major  tones,  two  minor 
tones,  and  two  major  semitones.  If  we  take  the  difference 
(-^)  between  the  major  3d  (f)  and  minor  3d  (|),  thus,  |-H-|=ff, 


ACOUSTICS  l6l 

we  get  the  minor  semitone.      We  may  notice  that  the  major 
and  minor  semitones  added  (X)  give  the  minor  tone  (f|  X  || 

1  0\ 

—  ^-)' 

i6.  We  may  sharp  or  flat  any  tone  by  multiplying  or  divid- 
ing its  ratio  either  by  ||  or  ||.  Using  §^  we  get  the  enhar- 
monic scale  of  2  I  notes  to  the  octave.  Using  |f  we  get  the 
enharmonic  scale  of  17  notes  to  the  octave.  Both  are  shown 
in  Fig.  S7- 


I 

11 

27 

1  25 

32 

4 
3 

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16 

8          6      125       9       15 

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M          M            M            M 

10      ^J        C        <-n 

<ji       1-1         0        0 
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CT>       O^      -^        CO       CD 

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76         5         18       225 
75        3        "¥"      T^S 

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0 
0 

0          M          M 
en         10         0 

4-        <-n         0 

^j 

10 

0 

t>5 

to 

0 
0^ 

? 

0 
0 

0 

LTK            O^        ^J           ^4 

0         0       ^4         t-n 
(0          0>       ^J          00 

CO 
^4 

Fig.  37. 

17.  The  ratios  are  also  given  decimally  for  more  conven- 
ient comparison,  from  which  it  is  seen  that  in  the  upper  scale 
E^  is  higher  than  Fl?,  and  Bjj:  than  Ct'  (the  sharps  and  flats  at 
these  points  overlapping) ;  and  that  in  the  lower  scale  the  re- 
sults are  all  overlapping  (If  being  greater  than  a  half-tone), 
and  the  tones  between  E  and  F  and  between  B  and  C  are 
eliminated  (|f  being  the  exact  interval  between  these  notes). 

18.  We  see  that  in  the  scale  of  nature  there  is  a  difference 
between  the  sharp  of  one  tone  and  the  flat  of  the  tone  next 
above,  and  also  that  there  is  a  great  lack  of  uniformity  in  the 
intervals  of  the  scale  as  shown  by  their  ratios. 

19.  If  these  intervals  were  equal,  transposing  or  modulat- 
ing to  different  keys  would  be  merely  a  shifting  of  the  scale 
(all  the  intervals  coinciding  in  each  change) ;  but  in  view  of 
the  lack  of  uniformity  shown  above,  transpo.sing  or  modulat- 
ing would  greatly  increase  the  number  of  tones  involved,  were 


l62  MUSICOLOGY 

all  the  keys  to  be  exact  (some  of  the  tones  would  coincide, 
but  the  majority  would  not). 

20.  The  violin  and  the  voice  are  capable  of  making  these 
changeable  intervals ;  but  the  inadaptability  of  the  scale  of 
nature  to  keyed  instruments  (as  the  piano)  is  very  apparent. 

21.  The  difTerent  methods  of  overcoming  this  difificulty  are 
called  Temperaments. 

EQUAL  TEMPERAMENT 

1.  The  scale  in  common  use  is  called  the  Equal  Tempera- 
ment Seale,  because  the  octave  is  divided  into  twelve  equal 
parts  called  semitones,  or  half-steps,  thus  conforming  to  the 
piano  keyboard,  which  contains  twelve  keys  (seven  white  and 
five  black)  to  each  octave. 

2.  Equal  Temperament  removes  the  distinction  between 
major  and  minor  tones  and  semitones,  and  also  between 
sharps  and  flats  (the  same  black  keys  being  used  for  both). 
The  enharmonic  scale  (a  scale  containing  intervals  smaller 
than  a  half- step)  is  thus  changed  to  the  chromatic  scale  (con- 
sisting of  half-steps),  and  the  enharmonic  change  (see  p.41  :  13) 
becomes  a  mere  change  of  name. 

3.  Since  the  octave  (2)  is  divided  into  12  equal  semitones 
we  may  correctly  regard  the  octave  either  as  12  semitones  {s) 
added,  thus,  j'  +  .f+j'+^  +  .J  +  J  +  ^  +  ^+^  +  J  +  J  +  .y 
=  12  .y,  or  as  12  semitones  multiplied,  thus,  .yX^Xi'XJX 
s  yi  s  y.  s  y.  s  'yi  s  y.  s  y.  s  y^  s  =^  s^'  \  for,  as  already  shown 
(p.  160 :  14),  multiplying  ratios  is  equivalent  to  adding  dif- 
ferences. We  must  bear  in  mind,  however,  that  the  semi- 
tones are  equal  only  in  tone  value,  but  not  in  number  of 
vibrations,  as  the  number  of  vibrations  in  any  interval  varies 
with  pitch. 

4.  Regarding  the  octave  as  12  semitones  multiplied  to- 
gether, therefore,  the  semitone  ratio  is  that  ratio  which  mul- 
tiplied by  itself    12   times  equals  the   octave  (2),  or  the    12th 

root  of  2  (]^2  =  1.05946  =  If  nearly);    but   for   the   sake    of 


ACOUSTICS 


163 


simplicity,   let  s   represent   the  ratio  of  the  semitone.      The 
chromatic  scale  will  then  be  represented  thus : 


S'-  =  2 


Vibration    values  — i       j'     s-       s'^      j-*     s^       s^      s''      s'^      s^       .f"^ 
Chromatic   scale  — C      Clt    D      D3    E      F      F$    G     G$    A      At     B      C 
Consecutive  ratios— i  X  s  X  s  X  s  X  s  X  s  X  s  X  s  X  s  X  s  X  s  X  s  X  J=2 

Dropping  the  sharp  tones,  the  diatonic  scale  will  be  thus: 
Vibration    values — i  .?'■'  s*     s^  j'  j-'  jH      j^'-'=:2 

Diatonic    scale       — C  D  E      F  G  A  B       C 

Consecutive  ratios— i      X      s^      X     s-  X  s      X       s'^     X     s'^      X      s'^  X  s  =2 

Fig.  :38. 

5.  We  see  that  each  vibration  value  is  the  product  of  all 
the  ratios  up  to  that  point  (see  p.   160 :  12). 

6.  As  the  semitone  (s)  is  the  common  unit  of  measure,  and 
as  the  exponent  of  j-  in  each  vibration  value  shows  the  num- 
ber of  times  s  is  used  as  a  ratio,  therefore  the  exponents  (as 
showing  the  number  of  semitones  contained)  are  the  measures 
of  the  intervals;  thus,  a  major  2d  is  2  semitones  (s^),  a  major 
3d  is  4  semitones  (s*),  a  perfect  4th  is  5  semitones  (/),  etc. 

7.  It  would  appear,  perhaps,  that  the  number  of  semitones 
should  be  expressed  by  coefficients,  thus,  2s,  4s,  5  j,  etc., 
instead  of  by  exponents,  thus,  J^  /,  /,  etc.  The  explana- 
tion of  this  is  that  the  semitone  is  expressed  in  the  form  of  a 
ratio,  and  not  in  the  form  of  a  difference. 

8.  Observe  that  (by  the  algebraic  rules  of  involution  and 
evolution)  we  multiply  by  adding  exponents,  divide  by  sub- 
tracting, raise  powers  by  multiplying,  and  extract  roots  by 
dividing  ;  thus,  s^  X  /  =  .y  ^  +  '^  =  / ;  /  h-  /  =  /  -  ^  =  / ;  (^s^y 
_.^x4=nj.i2.  ^  ^,12  __  ^12  H- G  _  ^,2 .  ^Q  j.j^^j.  ^j^g  exponents  of 
s  are  compared  (added,  subtracted,  etc.)  exactly  as  if  they 
were  coefficients. 

9.  The  exponents  of  s  may  also  be  called  logarithms.  A 
logarithm  of  any  number  is  the  exponent  of  the  power  to 
which  it  is  necessary  to  raise  a  fixed  number  (called  the  base)  to 
produce  the  given  number.      In  Fig.  38  the  log.  of  G  (perfect 


164  MUSICOLOGY 

5th)  is  7,  the  log.  of  E  (major  3d)  is  4,  etc.  (on  .y  as  a  base). 

10.  We  find  these  exponents,  or  logarithms,  sufficient  in  com- 
paring intervals  of  the  tempered  scale  with  each  other.  How- 
ever, in  comparing  the  tempered  scale  with  the  scale  of  nature, 
it  is  necessary  to  use  a   common  system  of  logarithms. 

11.  A  general  table  of  logarithms  (the  computation  of 
which  belongs  to  algebra)  gives  the  logarithms  of  all  numbers 
up  to  a  certain  point,  on  an  assumed  base  (usually  10),  giving 
also  the  decimals  involved  in  all  intermediate  numbers  (be- 
tween exact  powers),  usually  to  five  or  six  places  of  decimals, 
so  that  incommeasurable  numbers  are  thus  made  sufficiently 
commeasurable  for  comparison. 

1 2 .  The  value  of  a  logarithmic  table  consists  in  its  enabling  us 
to  multiply  by  adding,  divide  by  subtracting,  and  especially 
to  raise  powers  by  multiplying  and  extract  roots  by  dividing. 

13.  The  ordinary  process  in  using  the  table  consists  in 
first  finding  the  log.  of  the  given  number,  then  adding,  sub- 
tracting, multiplying,  or  dividing,  as  the  case  may  require; 
then  find  the  number  corresponding  to  the  resulting  log. 
However,  our  use  of  the  logarithms  here  is  only  as  a  basis  of 
comparison,  since  they  are  the  exponents  of  a  common  unit 
of  measure  (10),  and,  as  already  shown,  may  be  compared 
(added,  subtracted,  etc.)  as  if  coefficients. 

14.  The  log.  of  I  (10^  -^  10^  =  10^-'  =  lo*^*,  but  any  num- 
ber divided  by  itself  =  i,  therefore  10"—  i)  is  o,  and  the  log. 
of  10  (10'  =  10)  is  I  ;  therefore  the  log.  of  any  number  be- 
tween I  and  10  is  between  o  and  i,  and  therefore  wholly 
decimal.  As  the  interval  ratios  within  the  octave  are  be- 
tween I  and  2,  their  logs,  will  be  wholly  decimal. 

Logarithms  of  the  ratios]  

of   the    intervals   ot  tne  ;-         en  o  lo  ^i  to  ^  o 

Scale  of  Nature.  j       P,^^|J^^8tS 

Scale— C  D  E  F  G  A  B  C 


Logarithms  of  the  ratios  |       ^mOM-^M^o 
of   the    intervals   of   the  '  «  u>  4-  o-jo  o 

I  ^i  4^  uo    o  "^  -f»  <->> 

Tempered  Scale.  J 

Fig.  ;J9. 


ACOUSTICS  165 

15.  The  upper  logs,  in  Fig.  39  are  taken  from  a  loga- 
rithmic table  and  are  the  logs,  of  the  ratios  of  the  scale  of 
nature  (subtracting  the  log.  of  the  denominator  from  the  log. 
of  the  numerator  where  fractions  are  involved,  since  we  divide 
by  subtracting). 

16.  To  find  the  lower  logs,  we  first  divide  the  log,  of  the 
octave  (.30103)  by  12  (since  we  extract  roots  by  dividing), 
thus,  .30103  -^  12  =  .02509  =  log.  of  semitone;  now,  multi- 
plying (since  we  raise  powers  by  multiplying)  the  log.  of  the 
semitone  by  the  number  of  semitones  in  each  interval  of  the 
tempered  scale,  we  get  the  log.  corresponding  to  each. 

17.  The  log.  of  any  interval  not  given  in  Fig.  39  may  be 
found  by  taking  the  difference  between  the  logs,  of  those  let- 
ters between  which  it  is  found.  Thus,  between  E  and  G  is  a 
minor  3d,  the  log.  of  which  (.  17609— .09691)  is  .07918  in  the 
scale  of  nature,  and  (.  17560— .  10034,  or  .02509  X  3)  .07527 
in  the  tempered  scale. 

18.  We  have  now  a  basis  of  direct  comparison  between  the 
scale  of  nature  and  the  tempered  scale,  by  which  we  may  de- 
termine the  amount  of  deviation  of  each  interval  of  the  tem- 
pered scale  from  true  pitch.  Thus,  the  difference  between  the 
logs,  of  G  (perfect  5th)  is  .17609—  .17560=  .00049;  divid- 
ing by  the  log.  of  the  semitone  (.02509),  we  have  -gffg^  =  -gS" 
nearly :  therefore  the  perfect  5th  is  about  ^y  of  a  semitone 
flat  (the  log.  of  the  tempered  G  being  the  lower).  The  per- 
fect 4th  (inverted  perfect  5th)  will  of  course  be  -^j  of  a  semi- 
tone sharp.  This  deviation  from  true  pitch  is  scarcely  per- 
ceptible, and  therefore  the  4th  and  5th  are  classed  with  the 
octave  and  called  perfect  intervals. 

19.  Comparing  the  logs,  of  the  other  intervals,  we  would 
find  them  more  perceptibly  out  of  tune,  as  follows:  major  3d, 
about  Y  s  sharp;  minor  6th  (inverted  major  3d),  \s  flat; 
minor  3d,  ^s  flat;  major  6th  (inverted  minor  3d),  ^  s  sharp; 
major  7th,  \  s  sharp ;  minor  2d  (inverted  major  7th),  J  s  flat. 
The  whole  tone   has  a   double  value  in  the   scale   of  nature : 


1 66 


MUSICULOGY 


viajor  (ratio  |)  between  C  and  D,  F  and  G,  and  A  and  B 
(log,  .05115);  and  minor  (ratio  V)  betweeji  D  and  E,  and  G 
and  A  (log.  .04576).  In  the  tempered  scale  its  log.  is  .05017, 
which  is  about  g^j  s  flat  as  compared  with  the  major,  but  about 
\  s  sharp  as  compared  with  the  minor  tone.  The  major  2d  is 
therefore  3V  •$"  A^t  compared  with  the  major  tone,  or  \  s 
sharp  compared  with  the  minor  tone ;  the  inversion  (minor 
7th),  in  either  case,  being  the  opposite.  Our  ears  have  be- 
come so  accustomed  to  the  tempered  scale  that  we  do  not 
notice  the  slight  deviations  from  true  pitch. 

20.  By  means  of   the  fractions  thus  found  we  may  easily 

find  the  value  of  the  true  intervals 
in  semitones  ;  thus,  from  C  to  D  is 
a  major  2d  (2  .f)and  ^^s  flatter  than 
the  true  major  tone,  therefore  the 
true  major  2d  is  ^V  ^  sharper  (3^-^ 
=  .04)  or  2.04  s.  From  C  to  E 
is  a  major  3d  (4  s)  and  |  s  sharper 
than  the  true  major  3d,  there- 
fore the  true  major  3d  is  -^  j-  flatter 
(I  =  .  14)  or  4  J-  —  .  14  .y  =  3.86  .y. 

From  C  to  F  is  a  perfect  4th  (5  s)  and  ^V  ^  sharp  (51=  -02)  ; 
the  true  perfect  4th  is  therefore  5  ^—  .02  s  =  4.98  s.  From 
these  the  others  may  be  found ;  thus,  C  —  F  =  G  (see  table), 
F  +  E  =  A  and  G  -j-  E  =  B :  or,  the  difference  between  the 
values  of  any  two  intervals  is  the  value  of  the  interval  between  ; 
or,  the  sum  of  the  values  of  any  two  intervals  is  the  value  of 
the  interval  of  their  sum.  This  gives  the  values  to  within  one- 
hundredth  part  of  a  semitone,  which  is  much  beyond  the 
limit  of  sensible  error,  as  j\  s  is  regarded  as  the  sensible  limit. 

21.  In  this  way  we  get  a  more  definite   conception   of  the 
comparative  values  of  the  true  and  tempered  intervals. 


True 

Equal 

Intonation 

Temperament 

Semitones 

Semitones 

c 

12.00 

12.00 

b 

10.88 

11.00 

A 

8.84 

9.00 

G 

7.02 

7,00 

F 

4.98 

5.00 

E 

3.86 

4.00 

D 

2.04 

2.00 

C 

0 

0 

ACOUSTICS 


167 


CONSONANCE 
I.  Naturally,  the  closer  any  two  tones  are  related  the  more 
consonant  the  interval  between.  The  relationship  between 
any  two  tones  depends  in  the  first  place  upon  the  frequency 
of  the  blending  of  their  overtones.  In  Fig.  40  we  have  the 
different  consonant  intervals  within  the  octave  compared  in 
this  respect :  * 


n     ^e        •s 

•6         •! 

•6 

•  6        J«5 

•6 

•6 

♦8    ••^ 

•^      .1 

)        -5 

•5          ,-K 

•  5 

•  5           "4 

•5           'S 

•6           'S 

•6 

/\ 

•4             •2 

.4          '^ 

•  4 

•  3 

•^ 

•4 

#•5        '?«4 

J»5   'I'S 

c 

^     .3 
1     .3 

.3          '^ 

•*           .3 

•4 

\ 

•2 

-0 

.^      '2 

—♦2          Ol-      -•^2                    -^2 

-«-2 

-^2     '-                           -^2 

\^* 

o\ 

^^t 

— 

I'^i 

ZJr 

o\ 

N, 

.^'C?! 

o\ 

o\ 

01 

o\ 

01 

01 

C?i 

Octave  J 


Perfect  sth  ^      Perfect  4th  |       Major  3d  }        Major  6th  §         Minor  3d  |       Minor  6th 
Fig.  40. 


2.  Observe,  in  each  case,  that  the  frequency  of  the  blend- 
ing of  overtones  is  in  exact  accord  with  the  ratio  of  the  inter- 
val ;  thus,  in  the  octave  (ratio  f )  every  harmonic  of  the  upper 
note  blends  with  every  second  harmonic  of  the  lower  note,  in 
the  perfect  5th  (ratio  |)  every  second  harmonic  of  the  upper 
note  blends  with  every  third  harmonic  of  the  lower  note,  and 
similarly  as  to  the  other  intervals.  If  we  were  to  extend  each 
series  indefinitely  we  would  find  the  rule  continuing  to  hold  true. 

3.  These  blending  overtones  mutually  reinforce  each  other, 
the  amplitude  of  the  resulting  vibration  being  practically 
equal  to  the  sum  of  the  amplitudes  of  the  separate  vibrations; 
and  since  the  intensity  of  sound  is  proportional  to  the  square 
of  the  amplitude,  therefore  the  intensity  of  the  blending  over- 
tones is  equal,  not  to  the  sum,  but  to  the  square  of  the  sum 
of  the  separate  intensities.  This  gives  the  blending  overtones 
special  prominence. 


i68 


MUSICOLOGY 


cm 


4.  Observe,  in  Fig.  40,  that  in  each  interval  the  ratio  of 
the  number  of  harmonics  in  each  series  up  to  any  blending 
point  (reduced  if  necessary)  is  the  ratio  of  the  interval,  or  the 
vibration  numbers  of  any  two  blending  overtones  (put  into 
the  form  of  a  fraction  and  reduced  when  necessary)  gives  the 
ratio  of  the  interval ;  or,  if  Ave  regard  the  blending  overtones 
as  dividing  the  series  into  sections,  we  again  see  the  ratio  of 
the  interval  duplicated  in  each  section  in  the  ratio  of  the 
number  of  harmonics  on  each  side. 

5.  We  also  notice  that  the  remoteness  of  the  first  two  blend- 
ing overtones  is  also  a  measure  of  relationship. 

6.  Observe  that  in  the  octave  the  upper  note  with  all  its 
overtones  blends  with  overtones  of  the  lower  note,  the  two 
notes  thus  blending  into  one  without  adding  any  new  element, 

the  effect  being  merely  to  bright- 
en the  lower  note  by  strengthen- 
ing every  other  overtone  ;  hence 
octaves  are  regarded  as  replicates 
of  the  same  tone,  as  no  new 
element  is  involved. 

7.  Similarly  the  interval  of  a 
perfect  12th  (octave  -{-  perfect 
5th)  adds  no  new  element,  as 
every  harmonic  of  the  upper  note 
blends  with  every  third  har- 
monic of  the  lower  note.  The 
same  applies  to  any  interval 
which  corresponds  to  the  inter- 
val between  the  lower  note  and 
any  one  of  its  harmonics. 

8.  If  in  Fig.  40  we  were  to  ex- 
tend each  series  indefinitely,  we 

would  find  that  in  each  interval  the  blending  overtones  form  in 
themselves  a  harmonic  series,  as  their  vibration  numbers  are  1,2, 
3,  4,  etc.,  times  the  number  of  the  lowest  blending  ov^ertones. 


Major 
Cth 


5 — 

/ 

•  • 

(( 

>) 

\ 

J 

tJ 

^. 

> 

-y    (^ 

Fig.  n. 


ACOUSTICS  169 

9.  If  we  were  to  arrange  the  intervals  in  the  form  of  a 
downward  harmonic  series,  as  in  Fig.  41,  the  double  notes 
would  show  the  blending  overtones  of  each  separate  interval. 
(The  7th  harmonic,  which  is  omitted  in  Fig.  41,  as  it 
does  not  belong  to  the  scale,  may  be  included.)  By  ex- 
tending Fig.  41  sufficiently  we  may  include  also  the  dis- 
sonant intervals,  as  the  principle  holds  good  for  all  inter- 
vals. 

10.  The  relationship  between  any  two  tones  depends  in  the 
second  place  upon  the  frequency  of  the  blending  of  their  vibra- 
tions. In  the  octave  every  vibration  of  the  lower  tone  blends 
with  every  second  vibration  of  the  upper  tone  as  indicated  by 
the  ratio  f ;  thus,  [  '  I  '  I  '  I  •  (The  spaces  between  the 
marks  represent  the  comparative  duration  of  the  vibrations.) 
In  the  perfect  5th  (ratio  |)  every  second  vibration  of  the 
lower  tone  and  every  third  vibration  of  the  upper  tone 
blend  ;  thus,  J  '  ,  '  [  '  ,  '  [  .  Similarly,  in  the  perfect  4th 
(ratio  I)  every  third  vibration  of  the  lower  tone  blends  with 
every  fourth  vibration  of  the  upper  tone.  In  the  major  3d 
(ratio  I)  every  fourth  blends  with  every  fifth.  In  the  minor 
3d  (ratio  I)  every  fifth  blends  with  every  sixth.  In  the  major 
6th  (ratio  |)  every  third  and  fifth  blend.  In  the  minor  6th 
(ratio  I)  every  fifth  and  eighth  blend.  In  the  major  2d 
(ratio  I)  every  eighth  and  ninth  blend.  In  the  major  7th 
(ratio  V)  every  eighth  and  fifteenth  blend. 

11.  We  see  that  the  ratio  of  any  interval  is  duplicated,  both 
in  the  blending  of  the  overtones  and  in  the  blending  of  the 
vibrations,  and  therefore  the  whole  may  be  summed  up  in  the 
general  rule,  that  tJie  consonance  of  intervals  (and  therefore 
the  relationship  of  the  including  tones)  is  proportional  to  the 
simplicity  of  their  ratios.  We  naturally  conclude  that  sim- 
plicity is  the  essential  element  of  consonance,  and  that  the 
ratio  of  any  interval  is  its  characteristic  stamp  by  which  the 
same  interval  always  gives  the  same  impression  as  to  degree 
of  simplicity  regardless  of  pitch. 


I/O  MUSICOLOGY 

1 2 ,  The  blending  overtones  only  blend  perfectly  when  in  per- 
fect unison  ;  and  they  are  only  in  perfect  unison  when  the 
interval  between  the  generating  tones  is  just  (in  accord  with 
the  scale  of  nature).  But  in  the  tempered  scale  all  the  inter- 
vals (except  the  octave)  are  slightly  false,  and  hence  the 
blending  overtones  do  not  blend  perfectly,  but  sufificiently 
to  still  clearly  define  the  interval ;  and  the  effect,  if  percept- 
ible, is  only  a  sense  of  inaccuracy.  For,  as  already  seen  (p. 
i68:  4),  the  ratio  of  any  interval  is  duplicated,  not  only  in 
the  blending  vibrations  of  the  tones  forming  the  interval, 
but  also  in  the  frequency  of  blending  overtones,  in  the  har- 
monics as  a  whole  up  to  any  blending  point,  and  in  the  har- 
monics of  each  section  separately;  so  that  thus  the  true  ratio 
of  the  interval  is  clearly  defined,  even  though  the  interval 
itself  should  be  slightly  false.  There  could  be  no  sense  of 
inaccuracy  without  a  sense  of  the  true  interval  as  a  basis  of 
comparison, 

13.  Observe  that  all  the  consonant  intervals  of  the  scale  of 
nature  are  a  natural  selection  of  those  having  the  simplest 
ratios,  and  therefore  most  consonant. 

BEATS 

1.  Sound-waves  (p.  147  :  4)  increase  in  length  as  they  dim- 
inish in  number,  for  what  is  lacking  in  number  must  be  made 
up  in  length,  since  all  sounds  travel  the  same  distance  in  the 
same  time. 

2.  When  two  notes  of  different  pitch  are  sounded  together, 
the  longer  waves  of  the  lower  note  steadily  gain  on  the  shorter 
waves  of  the  upper  note;  and  the  gain  per  second  will  equal 
the  difference  in  the  number  of  waves  or  vibrations  per  second 
in  the  two  notes. 

3.  It  is  evident  that  the  reverse  phase  of  each  wave  gained 
must  be  met,  while  at  other  times  both  waves  move  in  the 
same  direction,  so  that  they  will  alternately  coincide  and  op- 
pose.     When  the)'  coincide  they  strengthen  each   other  and 


ACOUSTICS 


171 


increase  the  sound ;  when  they  oppose  they  counteract  each 
other  and  diminish  or  entirely  destroy  the  sound  ;  thus  pro- 
ducing an  alternate  swell  and  lull,  which  are  called  Beats. 

4.  The  number  of  beats  per  second  produced  by  any  two 
notes  sounding  together  is  equal  to  the  difference  in  the 
number  of  vibrations  per  second  of  the  two  notes. 

5.  Imperfect  Unison  Beats.  Tones  in  perfect  unison  do 
not  beat,  as  their  vibrations  coincide  ;  but  an  imperfect  unison 
is  marked  by  distinct  beats,  which  increase  in  rapidity  with 
increase  of  error.  This  is  very  important  in  tuning  instru- 
ments;  thus,  to  tune  two  notes  to  a  perfect  unison  it  is  only 
necessary  to  gradually  bring  them  together  till  they  cease  to 
beat. 

6.  Beats  which  are  due  to  imperfect  unison  are  called 
Imperfect  Unison  Beats. 

7.  Beats  may  occur  between  overtones,  which  are  therefore 
called  overtone  beats,  while  the  beats  between  the  prime  tones 
are  called  prime  beats.  Prime  beats  and  overtone  beats  often 
occur  at  the  same  time,  producing  more  or  less  confusion. 

8.  Though  on  the  one  hand  beats  increase  in  number  with 
increase  of  interval,  on  the  other  hand  they  rapidly  decrease 
in  strength  with  increase  of  interval  (the  reason  is  given 
under  "Sympathetic  Resonance");  therefore,  overtone 
beats  are  sometimes  stronger  than  the  prime  beats  by  reason 
of  a  closer  beating  interval. 

9.  Imperfect  Consonance  Beats.  Wide-apart  consonances 
(as  octaves,  perfect  5ths,  perfect  4ths,  etc.)  when  slightly 
false  also  produce  distinct  beats,  which  are  called  Imperfect 
Consonance  Beats,  and  which  also  increase  in  rapidity  with 
increase  of  error. 

10.  These  are  due  in  the  first  place  to  the  imperfect  uni- 
son beats  of  the  lowest  blending  overtones.  When  an  inter- 
val is  slightly  false,  all  the  blending  overtones  become  imper- 
fect unisons,  and  therefore  beat  because  of  their  imperfect 
blendins.      Of  course  the  beats  of  the  lowest  blending  over- 


lyZ  MUSICOLOGY 

tones  are  the  most  distinct  and  determine  the  number;  but 
these  are  strengthened  by  blending  beats  of  the  higher  blend- 
ing overtones. 

11.  The  number  of  beats  in  the  higher  blending  overtones 
are  respectively  2,  3,  4,  etc.,  times  the  number  in  the  lowest 
blending  overtones;  and  therefore  every  2d,  3d,  4th,  etc., 
beats  respectively  blend  with  the  lowest. 

12.  The  strength  of  these  imperfect  consonance  beats  de- 
creases with  decrease  in  consonance  of  interval,  by  reason  of 
the  increasing  remoteness  of  the  blending  overtones  (see 
Fig.  40). 

13.  It  has  been  shown  by  experiments  that  imperfect  con- 
sonance beats  occur  when  the  tones  are  simple  (without  over- 
tones);  but  it  has  also  been  shown  by  resonators  (tuned  jars 
which  detect  corresponding  disturbances  in  the  air)  that  these 
beats  are  formed  within,  and  not  outside  the  ear.  These, 
however,  correspond  in  number  and  therefore  blend  with  and 
strengthen  those  formed  by  the  blending  overtones  (when 
both  exist).  The  internal  beats  are  called  subjective,  as  they 
have  only  a  subjective  existence ;  while  the  external  beats  are 
called  objective,  as  they  have  an  objective  existence. 

14.  Helmholtz  explained  the  subjective  imperfect  conso- 
nance beats  on  the  theory  that  simple  toties  when  poiverful 
enough  produce  overtones  zvitJiin  the  ear  by  reason  of  the  pecu- 
liarly favorable  construction  of  the  ear-drum  and  connecting 
parts.  Koenig  claimed  that  in  his  experiments  no  overtones 
were  audible,  and  explained  the  beats  as  due  to  the  beating  of 
the  nearest  multiples  of  the  simple  tones.  As  these  multiples 
correspond  to  the  overtones,  the  result  is  the  same  as  if  we  as- 
sumed the  actual  existence  of  the  overtones.  It  makes  no 
difference,  so  far  as  the  sensation  (practical  fact)  is  concerned, 
whether  the  beats  are  objective  or  subjective;  and  therefore 
the  question  as  to  the  relative  objective  or  subjective  charac- 
ter of  the  beats  is  of  no  practical  (only  theoretical )  importance. 

15.  In  tuning  consonant  intervals,  as  in  tuning  unisons,  it 


ACOUSTICS  173 

IS  only  necessary  to  raise  or  lower  the  dissonant  note  (as  the 
case  may  require)  till  it  ceases  to  beat  or  (in  tuning  to  any 
degree  of  temperament)  till  the  number  of  beats  correspond- 
ing to  the  desired  degree  of  temperament  is  heard.  Beats 
thus  furnish  the  most  accurate  means  of  tuning. 

16.  Beat  Tones.  When  the  rapidity  of  the  beats  exceeds  16 
per  second,  they  begin  (owing  to  their  regular  periodic  recur- 
rence) to  form  a  tone,  which  is  therefore  called  the  beat  tone. 

17.  As  the  rapidity  increases  the  beating  effect  decreases, 
while  the  definiteness  and  pitch  of  the  beat  tone  increases ; 
but  during  the  transition  of  the  beats  into  a  tone,  both  beats 
and  tone  are  heard,  since  the  two  different  sensations  are  re- 
ceived by  different  parts  of  the  ear. 

18.  As  already  observed,  the  strength  of  beats,  and  there- 
fore of  the  beat  tone,  decreases  with  increase  of  interval. 
But  we  may  increase  the  number  of  beats  without  changing 
the  interval  by  taking  the  same  interval  at  a  higher  pitch, 
thus  raising  the  pitch  of  the  beat  tone ;  or,  on  the  other 
hand,  we  may  (ascending  the  scale)  narrow  the  interval  with- 
out changing  the  number  of  beats,  thus  increasing  the  loud- 
ness of  the  beat  tone  without  changing  its  pitch.  Also,  the 
closer  the  interval,  the  lower  the  pitch  of  the  beat  tone  as 
compared  with  its  generators,  and  therefore  the  more  easily 
heard  by  contrast.  Thus  we  see  that,  other  things  being 
equal,  beat  tones  increase  in  prominence  ascending  the  scale. 

19.  If  two  notes  (called  generators)  are  sounded  together, 
beats  or  beat  tones  will  exist  theoretically  between  all  the 
harmonics  of  the  two  series;  but  taking  into  account  the 
effect  of  distance  and  the  tendency  of  the  stronger  to  absorb 
the  weaker,  it  naturally  results  that  only  the  most  prominent 
have  audible  existence. 

20.  When  two  beat  tones  are  of  nearly  equal  intensity, 
both  may  be  heard  at  the  same  time  and  may  in  turn  beat  (if 
differing  slightly  in  pitch).  Also  beats  and  beat  tones,  due  to 
different  parts,  are  sometimes  heard  at  the  same  time. 


174 


MUSICOLOGY 


DIFFERENTIAL  AND    SUMMATIONAL  TONES 

1.  These,  like  beat  tones,  are  phenomena  resulting  from 
sounding  two  notes  together. 

2.  Differential  tones  have  a  vibration  frequency  equal  to 
the  difference  in  the  number  of  vibrations  of  their  generators, 
hence  the  name.  Summational  tones  have  a  vibration  fre- 
quency equal  to  the  sum  of  the  number  of  vibrations  of  their 
generators,  hence  the  name.  Summational  tones  are  there- 
fore always  higher  than  both  generators ;  while  differential 
tones  are  the  same  in  pitch  as  the  lower  generator  at  the 
octave  (the  upper  generator  being  double  the  lower),  but 
lower  than  both  generators  when  their  interval  is  less  than  an 
octave,  and  between  both  generators  when  their  interval  is 
greater  than  an  octave. 


7,  II,  and  13  are 
only  approxi- 
mate, as  their 
true  pitch  is  not 
exactly  shown 
bv  the  staff. 


f^Octave  2/j 

Per.  5">  %  P^.  ithy^iiajorSrd% 

Uinor^rde/^ 

*"" 

) 

y 

•  8 

^8 

f? 

) 

D  •   * 

^-m 

J 

•  5 

L-f5>s- 

L-^5-J 

L-1^^ 

-(©l 

~0l 

/7.^.«3 

03 

G3 

0^ 

•  3 

k> 

\^G2 

C?2 

•  2 

Jifa/W6"'-ys 

Jfi«or6'»% 

-^^\ 


-1        -•-1- 

FiG.  42. 


3.  Fig.  42  shows  the  differential  and  summational  tones 
for  the  consonant  intervals  within  the  octave.  They  of 
course  remain  relatively  the  same  for  each  interval,  regardless 
of  the  position  of  the  interval  on  the  staff.  Observe  that 
taking  the  difference  between  the  generator  numerals  gives 
the  differential  numeral,  and  adding  the  generator  numerals 
gives  the  summational  numeral. 

4.  Summational  tones  are  usually  weak  and  hence  are  of 
but  little  importance.      Differential  tones,  however,  are  often 


ACOUSTICS 


175 


distinctly  audible,  and  are  the  more  easily  heard    the  lower 
they  are  as  compared  to  their  generators. 

5.  Taking  the  difference  between  the  generators  themselves 
we  get  the  differential  of  the  first  order.  Taking  the  differ- 
ence between  the  differential  of  the  first  order  and  either 
generator  we  get  the  differentials  of  the  second  order.  Tak- 
ing the  difference  between  the  differentials  of  the  second 
order  and  either  generator,  or  the  differential  of  the  first 
order,  we  get  differentials  of  the  third  order.  And  so  on, 
theoretically,  till  the  harmonic  series  from  i  (root  of  interval) 
up  to  the  generators  is  complete,  which  may  then  be  con- 
tinued through  the  overtone  differentials  and  summationals' 
indefinitely. 

6.  Fig.  43  shows  the  harmonic  series  formed  by  the  dif- 
ferential tones  of  each  of  the  consonant  intervals  within  the 
octave ;  but  the  principle  applies  similarly  to  all  intervals. 
The  intervals  in  the  figure  are  arranged  in  their  natural  order 
in  the  harmonic  series,  and  hence  have  the  same  root ;  and 
the  harmonic  series  formed  in  each  case  is  the  same  as  far  as 
it  goes.  Otherwise,  the  intervals  would  have  different  roots 
and  the  differentials  would  form  different  harmonic  series. 


Octave  I 

n 

Per.  5th  i 

Per.  4th  1 

Major  3d  f 

Minor  3d  \ 

Major  6th 

i  Minor  6th 

§ 

) 

/ 

®%.T 

(( 

\ 

V 

J 

.      .  _. 

.^-ic 

■rf^ir 

«J 

-€^             -^A-:i              -^              -*^               -^i 

^■^' 

G^ 

GZ    'Z 

•  3 

•  3 

©3    'S 

•3 

(cJ- 

\ll/02 

02  m2 

•  2 

•  2 

•  2 

•2 

•  2 

-  -Oi  *i 

ist  (2  —  1  =  1) 
(Orders) 


-^h  -»-fc  —^1  —^1:  — ►!:  — ^> 

ist  (3-2=1)  ist  (4-3=1)  ist(s-4=i)  1st  (6-5=1)  ist  (5-3=2)  ist  (8-5  =3) 

2d  (3-.  =  2)  2d  (3II-)  .d  (^1-3)  ,d  (11-4)  .d  (3--)  2d  (^3-) 

3d  (5-3  =  2)  3d  (6-4  =  2)   3d  (5-1  =  4)   ,d/'8-2=6\ 
4th  (5-2=3)  3CH^3_3^J 

Fig.  43.  '''^8-1=7) 


176  MUSICOLOGY 

7.  The  different  orders  of  differentials  are  indicated  at  the 
bottom  of  Fig.  43.  It  will  be  seen  that  the  simpler  the  ratio 
of  any  interval  the  fewer  the  differentials  formed,  and  there- 
fore the  simpler  the  character  of  the  interval. 

8.  Fig.  44  shows  the  interval  of  a  major  3d  and  the  three 
harmonic  series  involved,  thus  showing  their  relation  to  each 
other.  The  middle  series  is  extended  up  to  the  summational 
tone  of  the  generators. 

9.  Observe  that  the  summational  of  the  generators  ex- 
presses also  a  differential  relation  indirectly  as  between  either 

£4»-  -•-25  generator  and  the  harmonics  of 

~^^  the    other,    as    shown  at   a\   or 

•15  as    between    the    harmonics    of 

each,  as   shown  at  b  and  c ;   or 

•  10 .       '  ■ 

X\  directly    as    between     the    har- 


-^7 


-♦6 


^^-      -•* 


monies  above    and    below  each 

^-5 •  blending  point,  as  shown  at  d\ 

and  is  thus,  in  a  sense,  a  focus- 

— ing   point  as  between    the    two 

75 "  harmonic  series. 

.        10.   Also,    the    summational 

—  of  the  overtones  taken  success- 

j-^  ively,  thus,    8+10=  18,   12  + 

io-(s-4)  =  9)  (  i2-r  8-5)  =  9    15  =  27,        16+20=36,        etc., 

i5-(io-4)  =  9r       ^       Ji6-(i2-.s)  =  9 


i 


^ 


2o-(i5-4)  =  Qf         ]2o-(i6-s)  =  9  are  in  like  manner  hifrher  focus- 
25  —  (20  —  4)  -  9 ;  ^  24  -  (20  -  5)  =  9  ^  o 

^^-  ^^-  *<^-   *<^-        ing  points ;    and  these  focusing 

,  ,  ,  „     ,         ,        X  ,^    o  N      points    taken    together   form  in 

(10-4) +  (  8-   s>=9         (i5-4)-(io-8  )=9    ^  ^ 

^'5-^^+("-^_°'=9     (2o-^8)-(i5-j2)=9  themselves   a    harmonic    series, 

d  thus,   9,    18,   27,   ^6,   etc.,    and 

2oV'2ol    24-13  =  9  therefore  harmonize  and    blend 

6/   \    -  1       25  —  16  =  9 

&c.'^  &c.  together,   forming    a  compound 

Fig.  44.  tone. 

II.  Also  the  differentials  of  the  harmonics  of  the  gener- 
ators taken  successively,  thus,  5  —  4=  1,  10  —  8  =  2,  15  — 
12  ^3,  20 —  16  =  4,  etc.,  form  a  harmonic  series  which  is 


ACOUSTICS  177 

always  built  on  the  first  differential  (which,  however,  is  not 
always  i),  and  if  extended  high  enough  will  include  the  sum- 
mationals.  Comparing  this  series  with  the  series  produced 
by  the  summationals,  we  see  that  they  stand  to  each  other, 
in  whole  or  in  part  or  in  any  similar  combination  of  parts,  in 
the  ratio  of  the  sum  and  difference  of  the  generators ;  so  that 
differential  and  summational  tones  are,  in  a  sense,  comple- 
ments of  each  other,  and  one  can  scarcely  exist  without  the 
other. 

12.  Observe,  again,  that  the  summational  of  the  generators 
is  the  exact  mean  between  the  first  overtones ;  and  each  sum- 
mational successively  is  the  exact  mean  between  the  even 
numbered  harmonics  of  the  generators.  In  each  case  the 
overtones  being  of  equal  rank  are  equal  in  strength,  suppos- 
ing the  generators  are  equal.  The  summationals  are  also 
exact  means  between  the  differentials  of  the  odd  numbered 
harmonics  of  one  side  and  the  generator  of  the  other;   thus, 

15  —  4=11        25  —  4  =  21 
mean  =  9  mean  ^=  18, 

12—  5  =7  20—5  =15 

etc.  Evidently  an  exact  mean  can  only  thus  be  indirectly 
formed  between  those  harmonics  having  an  uneven  differ- 
ence, since  all  elastic  bodies  must  vibrate  in  whole  numbers 
(exact  number  of  parts).  Other  interesting  relations  might 
also  be  pointed  out. 

13.  These  relations  are  general  and  apply  to  all  intervals, 
and  therefore  have  a  bearing  on  the  nature  of  differential  and 
summational  tones. 

14.  The  complemental  relations  between  differentials  and 
summationals  indicate  that  they  have  a  common  origin ;  and 
the  relations  pointed  out  indicate  that  they  are  the  resultant 
effect  of  the  influence  which  each  generator  exerts  on  the 
other. 

15.  Differential  tones  were  discovered  about  1745  by 
Sorge,    a   German   organist,    but   were  made  more  generally 


178  MUSICOLOGY 

known  by  the  Italian  violinist  Tartini,  and  are  often  called 
Tartini's  tones.  Summational  tones  were  discovered  by 
Helmholtz,  about  1854. 

16.  Helmholtz  showed  that  by  mathematical  theory,  when 
the  amplitude  of  the  vibrations  of  two  generators  sounding 
together  is  great  enough,  other  vibrations  are  also  produced 
having  a  frequency  corresponding  to  the  differential  and 
summational  tones;  and  thus  developed  the  theory  that  dif- 
ferential and  summational  tones  were  both  due  to  vibrations 
produced  by  the  combined  action  of  the  generators.  He  also 
showed  that  the  construction  of  the  drum  skin  of  the  ear 
and  connecting  parts  is  peculiarly  favorable  for  magnify- 
ing these  tones ;  so  that  they  are  thus  produced  in  the  ear 
even  when  they  are  not  sensibly  produced  in  the  air,  and 
hence  are  more  largely  subjective  than  objective. 

17.  Previous  to  Helmholtz's  theory  and  discovery  of  sum- 
mational tones,  differential  tones  were  generally  regarded  as 
due  to  beats,  and  therefore  identical  to  beat  tones. 

18.  Helmholtz  does  not  recognize  beat  tones,  evidently 
regarding  them  as  differential  tones.  Koenig  and  others  rec- 
ognize both  beat  tones  and  differential  tones,  but  as  differ- 
ing in  origin.  The  fact  that  audible  tones  sometimes  exist 
where  beat  tones  would  be  inaudible  by  reason  of  the  width 
of  the  interv^al,  would  also  indicate  that  these  tones  were  due 
to  some  other  cause  than  beats.  (Differential  tones  also  dim- 
inish somewhat  with  increase  of  interval.) 

19.  Beat  tones  and  differential  tones  both  have  a  vibration 
frequency  equal  to  the  difference  in  number  of  vibrations  of 
their  generators ;  therefore,  \vhere  they  both  exist  as  due  to 
the  same  generators,  they  coincide,  blend,  and  strengthen 
each  other.  This  would  appear  to  account  for  the  degree  of 
strength  which  neither  alone  would  sometimes  seem  to  war- 
rant;  especially  if  we  apply  the  principle  that  tlie  intoisity 
of  blending  tones  is  proportional  to  the  square  of  the  sum  of 
their  separate  intensities. 


ACOUSTICS 


179 


20.  Differential,  summational,  and  beat  tones  are  in  gen- 
eral called  combinational  tones,  because  produced  only  by  the 
combination  of  two  or  more  generating  tones.  They  are 
thus  distinguished  from  harmonics. 

DISSONANCE 

1.  Beats  bear  directly  on  the  subject  of  dissonance.  In- 
tervals increase  in  dissonance  with  number  of  beats  up  to 
about  33  per  second,  after  which  their  roughness  diminishes 
as  they  merge  more  and  more  into  the  beat  tone. 

2.  The  number  of  beats  in  the  major  2d  (whole  tone)  be- 
tween middle  C  of  256  vibrations  and  D  of  288  vibrations 
(288-256)  is  32,  and  hence  near  the  point  of  greatest  dis- 
sonance. If  taken  an  octave  higher  it  would  contain  64 
beats,  and  the  minor  2d  (semitone)  would  contain  32  beats. 
Or  if  taken  an  octave  lower  it  would  contain  16  beats,  and 
the  major  3d  (two  whole  tones)  would  contain  32  beats;  but 
here  the  increase  of  interval  counteracts  the  effect  of  the  beats, 
as  the  audible  limit  of  beats  is  about  a  minor  3d,  otherwise, 
any  interval  if  taken  low  enough  would  be  at  the  point  of 
greatest  dissonance.  However,  the  same  interval  increases 
perceptibly  in  roughness  descending  the  scale  till  the  point  of 
about  33  beats  is  reached,  and  vice  versa. 

3.  We  may  regard  the  interval  of  the  2d  (major  or  minor, 
according  to  pitch)  as  representing  about  the  maximum  de- 
gree of  dissonance. 

4.  Observe,  in  Fig.  40  (p.  167),  the  interval  of  a  2d  be- 
tween the  second  and  third  harmonics  in  the  perfect  4th  and 
major  and  minor  6ths,  and  between  the  third  and  fourth  har- 
monics in  the  perfect  5th  and  major  and  minor  3ds.  In  the 
perfect  5th,  however,  it  is  above  the  first  blending  overtones. 
We  observe,  also,  other  more  remote  dissonant  intervals. 

5.  We  see  that  all  intervals  (except  the  octave)  contain 
both  a  consonant  and  a  dissonant  element,  and  that  the  disso- 
nant element  increases  as  the  consonant  element  decreases,  so 


l80  MUSICOLOGY 

that  the  boundary  between  consonance  and  dissonance  is 
merely  a  question  of  predominance.  In  consonant  intervals 
the  consonant  element  predominates,  and  in  dissonant  inter- 
vals the  dissonant  element  predominates.  Fig.  40  shows 
only  those  within  the  octave  that  are  classed  as  conso- 
nant. 

6.  Where  more  than  two  generators  are  involved,  disso- 
nance is  due  in  a  large  measure  to  combinational  (especially 
differential)  tones.  In  Fig.  43  we  observe  that  where  only 
two  generators  are  involved  the  differentials  belong  to  the 
same  harmonic  series  with  the  generators  and  therefore  form 
consonant  intervals  with  them  (except  in  the  higher  parts, 
see  7,  in  the  case  of  the  minor  6th).  But  if  we  had  three  or 
more  instead  of  two  generators,  they  would  form  three  or 
more  different  intervals  with  each  other,  and  unless  these  in- 
tervals were  in  their  natural  harmonic  order  (as  only  in  the 
major  triad)  the  differentials  of  each  interval  would  form 
separate  harmonic  series,  which  would  form  a  greater  or  less 
number  of  dissonant  intervals  with  each  other  (producing 
beats);  but  naturally  the  more  harmonic  the  combination  of 
the  generators  the  fewer  the  dissonant  intervals  produced  by 
their  differentials. 

7.  If  Fig.  43  were  extended  to  include  the  dissonant  in- 
tervals, we  would  find  that  the  differentials  soon  begin  to 
form  dissonant  intervals  with  the  generators. 

8.  In  Fig.  42  we  sec  that  the  summational  tones  form 
dissonant  combinations  with  the  generators,  except  in  the 
octave,  perfect  5th,  and  major  6th  ;  but  as  summational  tones 
are  generally  weak,  if  audible  at  all,  they  may  usually  be  dis- 
regarded. 

SYMPATHETIC    RESONANCE 

I.  Sympathetic  Resonance  is  the  tendency  of  any  resonant 
body  to  vibrate  in  sympathy  when  a  note  is  sounded  near  it 
having  a  vibration  frequency  corresponding  to  its  own. 


ACOUSTICS  l8l 

2.  The  sympathetic  vibration  is  produced  by  the  repeated 
strokes  of  the  vibrations  of  the  sounding  body  being  trans- 
mitted to  the  sympathizing  body.  The  effect,  however,  is 
not  instantaneous,  as  the  first  stroke  in  itself  is  not  sufficient, 
but  must  be  magnified  by  repetition  ;  just  as  to  ring  a  very 
heavy  bell  requires  repeated  pulls  on  the  rope  at  regular, 
rightly  timed,  periodic  intervals  before  the  bell  begins  to 
ring.      The  principle  in  each  case  is  exactly  the  same. 

3.  Strings  are  very  sympathetic  when  attached  to  a  sound- 
ing-board (as  in  the  piano)  or  a  sounding-box  (as  in  the  violin 
or  guitar)  ;  but  the  strings  themselves  have  not  sufficient  sur- 
face to  receive  direct  very  much  of  the  force  of  the  trans- 
mitted strokes. 

4.  If  we  were  to  sing  a  note  into  a  piano  against  the  sound- 
ing-board (after  raising  the  dampers)  we  would  hear  the  note 
repeated  by  the  instrument  (the  note  should  be  sustained  for 
a  time  to  obtain  a  good  result).  The  vibrations  of  the  voice 
striking  the  sounding-board  are  transmitted  by  it  to  the 
strings;  and  those  strings  having  a  corresponding  vibration 
frequency  will  vibrate  in  sympathy.  If  the  note  sung  repre- 
sents a  compound  tone  (containing  overtones),  then  all  those 
strings  which  are  capable  of  sounding  the  whole  or  any  part 
of  the  compound  tone  will  sound  proportionately.  Thus  the 
piano  is,  in  a  measure,  capable  of  analyzing  the  compound 
tone. 

5.  Light  and  pliable  bodies,  such  as  strings,  membranes, 
and  enclosed  air,  being  easily  movable,  are  easily  set  into 
sympathetic  vibration,  which,  however,  as  readily  die  away 
when  the  cause  ceases.  On  the  other  hand,  heavy  or  rigid 
bodies,  as  bells,  tuning-forks,  etc.,  respond  less  readily,  and 
require  a  greater  number  and  more  accurately  timed  strokes 
and  therefore  longer  sustained  effort  to  produce  sympathetic 
vibrations,  which,  however,  die  away  as  slowly  as  they  are 
produced. 

6.  Easily  sympathetic  bodies  may  be  made  to  vibrate  by 


1 82  MUSICOLOGY 

tones  which  do  not  exactly  correspond  in  vibration  frequency 
(pitch),  but  the  ampHtude  of  the  vibrations  diminishes  as  the 
difference  in  pitch  increases;  and  naturally  the  more  easily 
sympathetic  a  body  is,  the  greater  the  limit  of  difference  in 
pitch  by  which  it  can  be  influenced. 

7.  The  ear  is  wholly  a  sympathetic  instrument.  The  end 
of  every  fibre  of  the  auditory  nerve  is  connected  with  small 
elastic  appendages  which  vibrate  sympathetically  with  the 
sound-waves.  Each  of  these  elastic  appendages  corresponds 
to  a  certain  pitch  of  tone,  but,  being  extremely  sympathetic, 
is  influenced  within  certain  limits  of  pitch  on  either  side ;  the 
sensible  limit  being  ordinarily  somewhat  more  than  a  semi- 
tone on  either  side,  or  together  about  a  minor  3d. 

8.  A  string  or  other  easily  sympathetic  body,  intermediate 
in  pitch  between  two  sounding  bodies  within  influencing  dis- 
tance, will  be  made  to  beat  by  being  set  in  sympathetic  vi- 
bration by  both  sounding  bodies  at  the  same  time ;  its  vibra- 
tions being  alternately  strong  (when  the  vibrations  of  the  two 
bodies  coincide)  and  weak  (when  they  oppose),  as  may  be 
seen  by  a  vibration  microscope. 

9.  This  illustrates  how  beats  are  produced  in  the  ear. 
Thus,  beats  are  produced  when  two  sounding  notes  are  near 
enough  to  each  other  in  pitch  so  that  the  same  clastic  appen- 
dages of  the  auditory  nerve  will  be  set  in  sympathetic  vibra- 
tion by  both  notes  at  the  same  time ;  the  combined  effect 
focusing,  as  it  were,  at  the  point  where  both  notes  produce 
equal  effect,  which  is  nearly  midway  between  (depending  on  the 
relative  strength  of  the  two  notes).  Naturally  the  combined 
effect  diminishes  as  the  distance  from  the  focus  increases,  by 
increase  of  interval.  This  explains  why  beats  decrease  in 
strength  with  increase  of  interval. 

10.  When  the  number  of  beats  is  sufUcient  to  produce  a 
beat  tone,  a  different  elastic  appendage,  corresponding  in 
vibration  frequency  to  the  number  of  beats,  is  brought  into 
sympathetic  vibration ;    thus,   the  beats   and  the   beat   tone, 


ACOUSTICS  183 

being  due  to   the  sympathetic   vibration  of  different    parts, 
may  be  heard  at  the  same  time. 

11.  We  noticed,  on  p.  181:4,  that  if  a  compound  tone 
were  sounded  into  a  piano,  those  strings  having  a  vibration 
frequency  corresponding  to  the  separate  simple  vibrations  of 
which  the  compound  sound-waves  are  composed  will  vibrate 
in  sympathy.  This  roughly  illustrates  the  principle  by  which 
the  ear  analyzes  compound  sound-waves ;  but,  unlike  the 
piano,  the  ear  is  capable  of  distinguishing  the  slightest  dif- 
ferences of  pitch,  and  its  sympathetic  sensitiveness  is  incom- 
parably greater. 

12.  The  principle  is  based  on  two  laws:  first,  Fourier's 
law,  the  substance  of  w^iich  (in  a  musical  sense)  is,  tJiat 
every  compound  vibration  is  the  sum  of  a  certain  mimber  of 
simple  vibrations,  and  is  tJierefore  capable  of  being  analyzed 
back  into  those  simple  vibrations;  second,  Ohm's  law,  the 
substance  of  which  is,  tJiat  the  ear  perceives  only  simple  vi- 
brations and  hence  does  not  recognize  the  compound  sound-wave 
as  a  single  ivhole,  but  o)ily  the  simple  elements  of  ivJiicJi  it  is 
composed. 


PART    FOURTH 


PRINCIPAL  SOURCES  OF  MUSICAL  SOUND 


OUTLINE 


■  Laws  of 
Vibration 


How 

excited 


^\ 


f  Rapidity  varies  inversely  as  the  length,  inversely 
J  as  the  thickness,  directly  as  the  square  root  of 
1  the  tension,  and  inversely  as  the  square  root  of 
I     the  weight. 

f  Loudness,  depending  on   amplitude  of 
rStrikine  vibration. 

\  Plucking  \  Pi'^.'^h'  depending  on  rapidity  of  vibra- 

t    owing      I  Quality,  depending  on   form  of  vibra- 
[     tion. 
Application— Piano,  Guitar,  Harp,  Violin,  etc. 
■  r.no-;tiiri;nai  i  Laws  of  Vibration— Same  as  in  transverse  vibrations. 
i/ihraHV.n<      -^  How  excited-By  rubbing  lengthwise  with  resined  cloth, 
^/ibrations       (  ^^^  ^^^^  -^^  music. 

Similar  to  a  stout  string  (rigidity   answering 
to  tension). 

Not  used  in  music. 
1^ Laws  of      (Rapidity   varies    directly    as    the 
J  v;vi-raf;r.n '^     thickuess  and  inversely   as   the 
1  Vibration  |     square  of  the  length. 
[Application-  Music-box,  etc. 
f  Supported  in  I   .      ,■     .■         n>  c     , 

I  the  middle       (  Application-I  uning-fork. 

1  Supported  at  /,,■.-         „ 
Nodal  Points  (  Application — Harmonicon,  etc. 

f  Law  of  Vibration— Rapidity  varies  inversely  as  the  length. 
(Series  corresponding  to  the  odd  figures,  i,  3, 


r  Transverse 

I  Vibrations    ! 


{ Both  ends  fixed 
I 
I 
.Transverse  I  Qne  end  fixed 

I 


Vibrations 


i  Both  ends  free 


.^    ,.      I  I  One  end  fixed    \     5 
Longitudina'  I  /go, 


etc. 


Vibrations 


Sound-wave  four  times  length  of  rod. 

Both  ends  fixed  |.  gound-wave  two  times  length  of  rod. 
Both  ends  free     I 


Q 


Air 
Columns 


Reeds 


Not  used  in  music, 
f  Vibrations— Longitudinal. 

Law  of  Vibration— Same  as  in  longitudinal  vibrations  of  rods. 

p-        )  Stopped  pipes— Analogous  to  rods  fixed  at  one  end. 

fipes-j  Open  pipes— Analogous  to  rods  free  at  both  ends. 
,  Application— Pipe  Organ,  Wood  Wind,  and  Brass  Wind  Instruments, 
i  Tongue-shaped— Application— Reed  Organs,  Clarinets,  etc. 
''  Mpinhrane    *  ^'Ps  in  playing  Brass  Instruments. 

Membrane    y  y^^^j^j  c^rds  in  Singing. 
Vibrations — Radial,  Circular. 


(Membranes-^  pi„^„  j  With  definite  pitch— Application— Ke 
,  I  i^iasb  ^  Without  definite  pitch— Application— D 


Kettledrum. 


rum, Tambourine,  etc. 


f  Vibrations— Radial,  Circular. 


„,   ^       It  f.jr-K,.„^;^r,  S  Other  things  being  equal,  rapidity  varies  directly 

Plates  ■!  Laws  of  Vibration  -^     ^^  ^j^^  thickness,  and  inversely  as  the  area. 

[  Application— Cymbals,  Gongs. 

\  (  By  strokes— Application— Church  Bells,  Clock  Chimes. 

!  How  excited  <     etc.  ,,     .     ,  ^, 

1  I  By  radial  friction— Application— Musical  Glasses. 

I  1  2i       4        6i  9 

tSeries-C         D        C         G»         D         etc. 


[Bells 


1 86  MUSICOLOGY 


STRINGS,  RODS    PIPES,  REEDS 

1.  Only  in  strings,  the  narrower  organ  pipes,  and  the  lon- 
gitudinal vibrations  of  rods  are  the  conditions  wholly  favor- 
able for  forming  the  natural  harmonic  series  with  vibrations 
proportional  to  the  simple  arithmetical  series  i,  2,  3,  4,  5, 
6,  etc.,  already  described. 

2.  Most  other  sounding  bodies  produce  overtones  which 
are  more  or  less  inharmonic  with  their  prime  tones. 

3.  Transverse  Vibrations  of  Rods.  In  the  case  of  rods 
fixed  at  both  ends,  we  have  conditions  somewhat  similar  to  a 
stout  string  (rigidity  answering  to  tension). 

4.  In  the  case  of  rods  fixed  at  one  end,  an  entirely  new  set 
of  conditions  prevails,  involving  new  laws  of  vibration,  which 
are  given  in  the  outline.  Ordinarily,  only  very  high  inhar- 
monic overtones  are  produced  when  the  rod  is  struck,  which 
do  not  blend  with  the  prime  and  soon  die  away ;  but  if  the 
rod  is  very  strongly  agitated,  some  of  the  lower  harmonic 
overtones  are  faintly  present.  These  last  are  present  only 
when  the  amplitude  of  vibration  is  sufficient  to  produce  a  sen- 
sible flexure  in  the  rod.  In  the  music-box  the  teeth  of  the 
music-comb  are  relatively  thin  as  compared  with  their  length, 
and  being  made  of  very  elastic  material,  they  admit  of  con- 
siderable amplitude  of  vibration,  and  therefore  tolerably  good 
musical  tones  are  produced.  (The  weights  on  the  teeth  also 
tend  to  develop  harmonic  overtones.) 

5.  In  the  case  of  a  rod  free  at  both  ends  and  supported  in 
the  middle,  each  half  is  of  the  nature  of  a  rod  fixed  at  one 
end.  A  tuning-fork  is  a  rod  of  this  kind  bent  on  itself  in  the 
middle  and  provided  at  this  point  with  a  stem  or  handle.  Its 
principal  use  is  as  a  standard  of  pitch,  for  which  it  is  specially 
fitted,  as  when  lightly  struck  and  held  close  to  the  ear  only 
the  prime  tone  is  audible.  The  high  inharmonic  overtones, 
however,    produce    a  sensible   tinkling   when    the   fork   is   first 


PRINCIPAL    SOURCES    OF    MUSICAL    SOUND  1 87 

struck,  but  soon  die  away ;   and  as  they  do   not  fuse  with  the 
prime  the  ear  easily  separates  them. 

6.  In  the  case  of  a  rod  free  at  both  ends  and  supported  at 
the  two  nodal  points,  the  section  between  the  points  of  sup- 
port will  be  of  the  nature  of  a  rod  fixed  at  both  ends,  and  the 
end  sections  will  be  of  the  nature  of  rods  fixed  at  one  end. 

7.  Longitudinal  Vibrations  of  Rods.  If  a  rod  fixed  at  one 
end  and  free  at  the  other  receive  a  blow  against  the  free  end 
lengthwise  of  the  rod  (or  rubbed  lengthwise  with  a  resined 
cloth  or  the  moistened  finger),  the  impulse  runs  along  the 
rod  to  the  fixed  end  and  is  reflected  back  to  the  free  end,  but 
is  now  as  a  pulling  force  instead  of  a  compressing  force;  and 
in  this  phase  it  runs  along  the  rod  to  the  fixed  end  and  back 
to  the  free  end,  when  it  is  again  in  its  original  phase,  thus 
making  a  complete  vibration,  since  from  like  phase  to  like 
phase  is  a  complete  vibration :  hence,  the  time  of  vibration  is 
the  time  required  for  the  impulse  to  travel  four  times  over  the 
rod,  and  the  wave-length  is  four  times  the  length  of  the  rod. 

8.  If  the  rod  be  fixed  at  both  ends,  the  effect  will  be  as  if 
each  half  were  fixed  at  one  end  and  the  middle  of  the  rod 
were  the  free  end  of  each  half;   thus,  a  rod  fixed  at  both  ends 

vibrates  in  the  same   time  as  a   rod    ^, — ^ 

half    the    length    fixed  at    one    end ; 


therefore,  the  time  of  vibration  is  the    ^ 

time  required  for  the  impulse  to  travel     Ci         " — l-         — 1 — =^ — l 

twice    over    the   rod,  and  the  wave-  ''"   ^' 

length  is  twice  the  length  of  the  rod. 

9.   Fig.  45   shows  how  a  rod  fixed  at  both  ends 

will  break  up  into  segments  producing  overtones : 

a  shows  the  rod  vibrating  as  a  whole  ;   d,  in  halves ; 

r,    in   3ds.      In    like    manner   the    rod   will    divide 

_      7      -^    into   4ths,    5ths,   6ths,   etc.  ;    the   harmonic    series 
CO      o     c    ,    .  ' 

Fig.  46.        bemg  the  same  as  ni  strmgs. 

10.    Fig.  46  shows  how  a  rod  fixed  at  one  end  will  break  up 

into  segments:   a  shows  the  rod  vibrating  as  a  whole  ;   l^  shows 


1 88  MUSICOLOGY 

it  vibrating  in  two  segments,  the  node  forming  at  one-third 
the  whole  length  from  the  free  end,  the  upper  segment  vibra- 
ting as  a  rod  fixed  at  one  end  and  the  lower  segment  as  a  rod 
twice  as  long  fixed  at  both  ends,  both  therefore  vibrating  in 
the  same  time  (in  all  cases,  the  rod  must  divide  so  that  each 
segment  will  vibrate  in  the  same  time) ;  c  shows  the  rod  vibra- 
ting in  three  segments,  the  upper  segment  being  half  the 
length  of  the  others — the  upper  node,  therefore,  will  be  one- 
fifth  the  Avhole  length  from  the  end,  so  that  each  segment 
may  vibrate  in  the  same  time, 

1 1.  Since  a  rod  fixed  at  one  end  vibrates  as  a  rod  twice  the 
length  fixed  at  both  ends,  therefore,  if  we  regard  b  as  the 
lower  half,  the  whole  rod  would  be  divided  into  three  seg- 
ments; or,  in  the  case  oi  c,  into  five  segments.  So  we  see 
that  the  harmonics  of  a  rod  fixed  at  one  end  will  be  in  the 
proportion  of  the  odd  numbers,  i,  3,  5,  etc.  ;  the  overtones 
corresponding  to  the  even  numbers  not  being  able  to  form. 
„  — ^     ^    "g^—  12.    Fig.  47  shows  how  a  rod  free 

at  both  ends  will  break  up  into  seg- 
^  '  '  ments :    a    shows    the   fundamental 

C^     !   ~^ — ! — '^^^^ — ^^^=^      vibration  of  the  rod,  which  forms  a 
^'"''  '*'■  node   in   the    middle — each   end    vi- 

brating as  a  rod  fixed  at  one  end  ;  b  shows  the  rod  with  two 
nodes,  each  one-fourth  of  the  whole  length  from  the  end,  the 
end  segments  vibrating  as  rods  fixed  at  one  end  and  the 
middle  segment  as  a  rod  twice  the  length  fixed  at  both  ends, 
so  that  the  time  of  each  is  the  same ;  c  shows  the  rod  with 
three  nodes,  the  outer  segments  being  half  the  length  of  the 
others  but  vibrating  as  rods  fixed  at  one  end,  while  the  inner 
segments  vibrate  as  rods  fixed  at  both  ends,  so  that  the  time 
of  each  is  the  same. 

13.  The  arrows  in  each  figure  represent  one  phase  of  the 
vibration ;  reversing  the  arrows  would  represent  the  opposite 
phase.  In  each  figure  we  must  conceive  of  the  rod  vibrating 
as  a,  b,  c,  etc.,  at  the  same  time  to  form  a  conception  of  the 


PRINCIPAL    SOURCES    OF    MUSICAL    SOUND  1 89 

compound  nature  of  the  vibrations;  as  the  rod  not  only 
vibrates  as  a  whole,  but  also  in  segments  forming  overtones 
with  the  prime,  similarly  as  already  explained  in  strings,  ex- 
cept that  the  vibrations  are  longitudinal  instead  of  transverse. 

14.  Pipes.  A  pipe  closed  at  one  end  (the  mouth  is  always 
an  open  end)  contains  a  rod  of  air  fixed  at  one  end,  which 
therefore  vibrates  like  a  rod  fixed  at  one  end  ;  and,  like  the  rod, 
its  harmonic  series  will  be  in  the  proportion  of  the  odd  num- 
bers, I,  3,  5,  etc.,  the  effect  of  which  is  to  make  the  tone 
more  dull  and  hollow. 

15.  A  pipe  open  at  both  ends  contains  a  rod  of  air  free  at 
both  ends,  which  therefore  vibrates  like  a  rod  free  at  both 
ends. 

16.  The  prime  note  of  the  pipe  closed  at  one  end  will  be 
an  octave  lower  than  the  same  pipe  open  at  both  ends ;  as 
the  open  pipe,  like  the  rod  free  at  both  ends,  contains  a  node 
in  the  middle  in  sounding  its  prime  note  and  therefore  sounds 
the  same  note  as  a  closed  pipe  half  the  length. 

17.  The  pitch  of  pipes  is  mainly  a  question  of  length. 
The  width,  however,  afTects  the  quality  of  tone,  the  promi- 
nence of  the  overtones  diminishing  as  the  width  increases;  so 
that  wide  pipes  produce  hollow  tones,  while  narrow  pipes 
produce  bright,  penetrating  tones. 

18.  In  cylindrical  pipes  only  every  other  overtone  is  prom- 
inent, producing  a  mellow  tone.  Flutes  and  clarinets  are 
made  on  this  principle. 

19.  Conical  pipes  tend  to  make  the  higher  available  over- 
tones relatively  more  prominent,  thus  producing  a  more  pene- 
trating tone.  Oboes  and  bassoons  are  made  on  this  prin- 
ciple. 

20.  A  peculiarity  of  the  pipe,  or  tube,  is  that  its  prime 
tone  can  be  made  to  jump  from  one  harmonic  to  the  next 
successively  by  gradually  increasing  the  blowing  force,  so 
that  any  harmonic  can  be  produced  as  the  prime  by  blowing 
with    the    proper  force.      The    narrower  the    pipe  the   more 


190  MUSICOLOGY 

prominent  the  overtones,  and  the  more  readily  will  the  prime 
jump  from  one  to  another.  This  principle  is  employed  more 
or  less  in  the  various  wood-wind  and  brass-wind  (especially 
brass-wind)  instruments. 

21.  Brass  instruments  without  keys  are  called  natural  in- 
struments, and  are  capable  of  producing  only  the  tones  cor- 
responding to  their  natural  harmonics. 

22.  Keys  on  brass  instruments  are  for  the  purpose  of 
lengthening  the  tube,  thus  lowering  the  pitch  of  the  instru- 
ment, thereby  producing  lower  harmonic  series.  In  three- 
keyed  instruments,  the  first  key  lowers  the  pitch  a  tone ;  the 
second,  a  semitone;  the  third,  a  minor  3d;  the  first  and 
second  together,  also  a  minor  3d  ;  the  second  and  third,  a 
major  3d  ;  the  first  and  third,  a  perfect  4th  ;  and  all  three  to- 
gether, an  augmented  4th  or  diminished  5th;  thus  lowering 
the  pitch  from  one  to  six  semitones.  By  a  proper  selection 
from  among  the  tones  of  the  several  harmonic  series  thus  at 
command,  the  entire  chromatic  scale  may  be  produced. 

23.  Other  brass  instruments  (sliding  trombones)  have  slid- 
ing sections  which  are  drawn  into  six  different  positions,  each 
successively  lowering  the  pitch  of  the  instrument  a  semi- 
tone. 

24.  Tone  is  produced  on  most  of  the  brass  instruments  by 
the  vibration  of  the  player's  lips  as  he  blows  into  the  cup- 
shaped  mouth-piece ;  the  pitcJi  depending  on  the  length  of 
the  tube,  the  force  of  blowing,  and  the  rigidity  or  laxity  of 
the  lips ;  the  quality  depending  on  the  width  of  the  tube  and 
the  shape  of  the  mouthpiece  (the  narrower  the  tube  and  the 
shallower  the  mouthpiece  the  brighter  and  more  ringing  the 
tone,  and  vice  versa),  and  also  on  the  size  of  the  bell  and 
shape  of  the  lips. 

25.  In  certain  instruments  of  the  flute  order,  the  tone  is 
produced  by  blowing  at  a  certain  angle  and  with  a  certain 
force  against  the  edge  of  the  mouth  of  the  tube. 

26.  In    the    organ    pipe    and    some    other    instruments,   a 


PRINCIPAL  SOURCES  OK  MUSICAL  SOUND  I9I 

mouthpiece  directs  the  current  of  air  at  the  proper  angle 
against  the  edge  of  the  tube ;  and  still  other  instruments 
have  reeds  in  their  mouthpieces  to  produce  the  necessary 
vibration.  Clarinets  have  single-reed  mouthpieces ;  oboes, 
bassoons,  and  English  horns  have  double-reed  mouthpieces. 
So  we  see  that  the  column  of  air  enclosed  in  a  pipe  is  set  in 
motion  by  various  contrivances,  but  on  two  general  princi- 
ples :  first,  direct,  by  forcing  the  air  against  the  edge  of  the  pipe 
at  a  certain  angle,  thus  producing  a  flutter  of  air  at  one  end ; 
second,  indirect,  by  communicating  the  vibration  of  some 
other  substance  (as  the  lips  or  reeds)  to  the  air  of  the  tube. 

27.  Resonance  Boxes  and  Sounding-boards  are  not  strictly 
sources  of  musical  sound,  as  they  merely  strengthen  but  do 
not  originate  sound.  They  are  used  mainly  to  strengthen 
the  sound  of  vibrating  strings,  as  in  the  violin,  piano,  etc. 
The  resonance  box  (with  the  enclosed  air)  or  sounding-board 
vibrating  in  unison  with  the  strings  strengthens  the  sound  by 
reason  of  its  greater  vibrating  volume.  The  vibrations  of 
the  strings  are  transmitted  to  the  box  or  board,  which  in 
turn  transmits  them  to  the  atmosphere.  The  direct  vibra- 
tions of  strings  can  be  heard  only  a  very  short  distance,  so 
that  the  tones  of  the  violin,  piano,  etc.,  are  due  mainly  to  the 
resonance  box  or  sounding-board. 

28.  Reeds.  A  reed  in  musical  instruments  is  a  thin  tongue 
of  wood  or  metal  fastened  at  one  end  to  a  slotted  plate  in 
such  a  manner  that  the  tongue  will  vibrate  either  within  or 
against  the  slot  (the  former  called  free,  the  latter  beating 
reeds). 

29.  In  free  reeds  the  tongue  exactly  fits  the  slot,  except 
that  it  is  enough  smaller  to  vibrate  freely  within  the  slot.  In 
beating  reeds,  either  the  tongue  or  the  slotted  surface  is 
slightly  curved  to  permit  the  necessary  play. 

30.  In  either  case,  if  a  current  of  air  be  passed  through  the 
slot,  the  tongue  will  be  pressed  alternately  into  or  against 
the  slot  by  the  pressure  of  the  air  and  out  by  the  reaction  of 


ig2  MUSICOLOGY 

its  own  elasticity,  so  that  the  air  -".vill  pass  through  in  inter- 
mittent puffs.  The  end  of  the  tongue  is  thinned  down  to  a 
feather  edge  for  high  notes  and  sometimes  weighted  for  low 
notes,  so  that  the  rapidity  of  vibration  (and  therefore  pitch 
of  tone)  depends  on  the  weight,  length,  and  thickness  of  the 
tongue.  The  tone  produced  is  due  to  the  intermittent  puffs 
of  air,  and  not  to  the  vibrations  of  the  tongue,  as  might  be 
supposed ;  the  reed  being  merely  a  mechanical  contrivance 
for  rapidly  opening  and  closing  the  passage-way  of  the  air, 

31.  Owing  to  the  varying  size  and  shape  of  the  opening, 
during  each  opening  and  closing,  the  pulses  of  air  produced 
are  compound  in  form,  resulting  in  compound  tones  with 
prominent  overtones.  When  reeds  are  used  with  tubes,  the 
quality  is  much  improved  by  the  sympathetic  resonance  of 
the  tube,  which  strengthens  the  tones  corresponding  to  its 
own  proper  tones  and  damps  the  others. 

32.  The  lips  in  blowing  brass  instruments  and  the  vocal 
cords  in  singing  are  classed  with  reeds,  as  they  are  made  to 
vibrate  in  a  similar  manner  by  a  current  of  air. 

MEMBRANES,   PLATES,   BELLS 

1.  These  may  be  classed  together  under  the  head  of  sur- 
face vibrations.  The  preceding  division  involved  only  a 
lengthwise  dimension,  and  the  vibrations  were  confined  to 
this  one  dimension;  but  surfaces  admit  of  vibrations  in  all 
lateral  directions,  which  evidently  makes  the  analysis,  as 
well  as  the  series  of  overtones  produced,  more  complex. 

2.  Membranes  and  Plates  being  similar  in  some  respects 
may  be  considered  together.  Their  vibrations  may  be  di- 
vided into  two  general  classes,  radial  and  circular,  which  are 
usually  more  or  less  combined. 

3.  Fig.  48  may  represent  either  a  circular  plate,  or  disk,  or 
a  stretched  membrane  as  a  drumhead.  The  circles  within 
represent  the  nodal  lines  of  the  radial  vibrations,  and  the 
diameters  represent  the  nodal  lines  of  the  circular  vibrations. 


J-KIXCIFAL    SOURCES    OF    MUSICAL    SOUND 


193 


4,    If  a  drumhead  be   struck    exactly  in   the   center,  vibra- 
tions will  radiate   from  the  center  similarly  as  when  a  pebble 


Fig.  48. 

is  dropped  into  the  center  of  a  circular  pool  of  water;  and 
when  these  radial  vibrations  break  up  into  nodes  they  form 
nodal  circles  about  the  center  (the  signs  +  and  —  showing 
opposite  phases  of  vibration) :  a  represents  the  vibrations  of 
the  prime  tone ;  and  b  and  r,  the  vibrations  of  overtones. 
If  the  stroke  is  at  one  side  of  the  center  of  the  drumhead, 
other  vibrations  will  also  be  produced,  which  tend  to  circle 
around  the  center  and  form  nodal  lines  through  the  center,  as 
at  d  and  c.  The  vibrations  circle  both  ways  from  the  point 
of  the  stroke  and  meet  at  the  opposite  side,  thus  always 
dividing  the  membrane  into  an  even  number  of  segments  by 
nodal  diameters.  Both  systems  of  vibrations  usually  exist 
together,  as  at ./. 

5.  The  overtones  produced  by  either  system  are  mainly  in- 
harmonic w^ith  the  prime  tone;  and  when  the  two  systems 
are  combined,  many  inharmonic  overtones  of  nearly  the  same 
pitch  are  produced. 

6.  If  we   suppose   Fig.  48  to  represent  a  circular  plate,  or 


194  MUSICOLOGY 

disk,  then  if  this  disk  is  supported  at  the  rim,  the  conditions 
are  somewhat  similar  to  the  drumhead  (rigidity  answering  to 
tension) ;  and  if  struck  at  the  center,  radial  vibrations  with 
nodal  circles  will  be  produced  ;  but  if  the  disk  is  supported  at 
the  center  (or  suspended  from  the  rim)  and  struck  at  the  rim, 
the  conditions  are  reversed,  and  circular  vibrations  with  nodal 
diameters  will  be  produced. 

7.  Under  all  other  conditions,  between  these  theoretical 
extremes,  both  systems  of  vibrations  will  be  combined ;  but, 
in  general,  the  nearer  the  stroke  is  at  the  center  the  more  the 
radial  vibrations  predominate,  and  the  nearer  the  stroke  is  at 
the  rim  the  more  the  circular  vibrations  predominate. 

8.  The  overtones  produced  by  plates  differ  from  those  of 
membranes,  but  are  also  mainly  inharmonic  with  the  prime, 
and  involve  also  inharmonic  overtones  of  nearly  the  same 
pitch  (producing  an  empty  tin-kettle  quality)  where  the  two 
systems  of  vibrations  are  combined.  In  both  cases  the 
series  varies  with  varying  conditions  and  cannot  in  any  case 
be  represented  accurately.  (The  series  produced  by  the  cir- 
cular vibrations  of  plates,  as  in  the  gong,  is  similar  to  that  of 
bells,  p.   195  :  15.) 

9.  Fig.  48  represents  only  a  few  of  the  more  simple  forms 
of  vibration  of  the  simplest  and  most  symmetrical  form  of 
membranes  or  plates.  Any  more  irregular  form  of  mem- 
branes or  plates  would  evidently  give  only  more  complex  re- 
sults. I-ong  narrow  plates  would  more  properly  be  classed 
as  rods. 

10.  Plates  are  not  much  used  in  music.  Cymbals,  how- 
ever, are  used  in  military  bands  to  mark  time. 

11.  Membranes  as  used  in  music  are  of  two  classes:  those 
w^ith  definite  pitch,  as  in  the  kettledrums;  and  those  with- 
out definite  pitch,  as  in  the  bass  and  tenor  drums  and  the 
tambourine. 

12.  The  kettledrum  consists  of  a  kettle-shaped  shell  of 
thin  brass   or  copper,  over   the  mouth  of   which  is  stretched 


PRINCIPAL   SOURCES    OF    MUSICAL   SOUND  I95 

the  membrane  called  the  drumhead;  the. tone  being  rein- 
forced by  the  associated  air-chamber  thus  formed.  Kettle- 
drums may  be  tuned  within  the  compass  of  a  perfect  5th  by 
varying  the  tension  of  the  membrane.  They  are  generally 
used  in  pairs  of  unequal  sizes,  and  are  usually  tuned  to  give 
the  tonic  and  dominant.  Sometimes  three  are  used  to  give 
the  tonic,  dominant,  and  sub-dominant. 

13.  The  bass  drum,  the  tenor  drum  (called  also  side  drum, 
snare  drum,  and  military  drum),  and  the  tambourine  are 
used  chiefly  to  mark  the  rhythm  of  music  without  any  regard 
to  their  pitch ;  they  also  excite  the  ear  to  a  more  acute  ap- 
preciation of  other  sounds.  The  quality  of  tone  and  also  the 
pitch,  to  a  limited  extent,  may  be  altered  by  changing  the 
tension  of  the  drumhead.  The  pitch  depends  mainly  on  the 
size;  thus,  the  bass  drum,  being  larger,  gives  a  lower  pitch 
than  the  tenor  drum. 

14.  Bells.  A  Bell  is  practically  a  bowl-shaped  plate;  the 
shape  tending  to  eliminate  the  radial  vibrations  and  empha- 
size the  circular  vibrations.  In  large  bells  the  rim  against 
which  the  clapper  strikes  is  thickened  (called  the  sound-bow). 

15.  In  sounding  its  deepest  tone  a  bell  divides  into  four 
segments.  It  may  also  divide  into  any  even  number  of  seg- 
ments. In  bells  of  uniform  thickness  throughout,  the  vi- 
bration numbers  of  the  series  produced  are  proportional  to 
the  squares  of  the  even  numbers,  4,  6,  8,  10,  12,  etc.,  into 
which  the  bell  divides  itself.  The  squares  of  these  numbers 
would  be  16,  36,  64,  100,  144,  etc.  Reducing  by  dividing 
through  by  16,  we  get  i,  21,  4,  6^,  9,  etc.,  which  would 
produce  the  following  series  : 

I  2J  4  6i  9 

C^,    D,,     C3,      G«,  D,,  etc. 
(The   lower  figures    indicate   the  octave    within     which     the 
tone   is  found.)     From   Fig.  33   (p.    157)    we   find    that    the 
vibration     number    4    corresponds  to  Q;    and    9,     to    D^; 
their  ratio,  |  or  2i,  to  D^.      Sharping  G^  by  multiplying  its 


196  MUSICOLOGY 

vibration  number  (6)  by  |f  (the  minor  semitone),  we  get  6|. 
None  of  these  tones  are  harmonic  to  C  except  C3. 

16.  Overtones  are  more  prominent  in  thin  than  in  thick 
bells ;  also  in  shallow  than  in  deep  bells.  The  deeper  tones 
may  be  made  mutually  more  harmonic  by  giving  the  bell  a 
certain  empirical  shape.  The  tone  also  depends  on  the 
elasticity  of  the  material  and  the  thickness  of  the  sound-bow. 
The  body  of  the  bell  gives  a  deeper  tone  than  the  sound- 
bow,  but  not  so  loud. 

17.  The  principal  elements,  then,  which  determine  the 
tone  of  a  bell,  are  weight,  size,  shape,  thickness  of  the 
sound-bow,  and  elasticity  of  the  material.  If  the  bell  is  not 
cast  perfectly  homogeneous  and  symmetrical,  irregularities 
naturally  result. 

18.  Musical  glasses  may  be  classed  with  bells,  as  being 
similar  in  shape,  though  excited  by  friction  instead  of  by 
strokes.  When  a  bell  is  excited  by  a  stroke  the  vibrations 
are  necessarily  transverse  in  character;  but  in  musical  glasses 
the  vibrations  are  produced  by  passing  the  moistened  fingers 
around  the  rim  of  the  glass,  and  therefore,  from  the  direction 
of  the  motion  producing  the  vibrations,  we  see  that  they 
must  be  longitudinal  (in  the  direction  of  the  rim)  in  character. 
We  would  also  judge,  from  the  musical  quality  of  the  tone, 
that  the  series  produced  is  harmonic. 


i 


PART    FIFTH 


APPENDIX 


HISTORY  OF  THE  DIATONIC  SCALE 

1.  The  earliest  stages  of  music  were  naturally  such  as  we 
still  find  among  savage  tribes,  consisting  of  a  few  sounds  dif- 
fering in  pitch  but  without  any  system  as  to  their  relations 
and  the  mode  of  using  them.  But  in  the  growth  of  music 
from  this  rude  state  different  systems  have  developed,  many 
of  which,  besides  our  own,  are  still  in  use,  especially  in  orien- 
tal countries. 

2.  Most  of  these  systems  recognize  the  octave,  and  some 
also  the  4th  and  5th.  The  Chinese,  Japanese,  and  some  other 
countries  use  chiefly  a  scale  of  five  tones,  called  the  Penta- 
tonic  Scale.  The  dividing  points  vary,  however.  The  Hin- 
doos divided  the  octave  theoretically  into  twenty-two  parts, 
but  practically  into  seven  degrees  changeable  into  distinct 
modes  in  a  manner  somewhat  similar  to  our  own.  The  Per- 
sians divided  the  octave  into  twenty-four  parts,  each  corre- 
sponding to  half  our  semitone.  The  Arabs  divided  the  octave 
into  sixteen  or  seventeen  parts  (according  to  different  author- 
ities).     Other  systems  might  also  be  mentioned. 

3.  As  to  the  ancient  music  of  the  Egyptians,  nothing  is 
clearly  known  regarding  the  scale  except  that  the  octave  was 
largely  subdivided.  The  music  of  the  Chaldeans,  Babyloni- 
ans, Assyrians,  Phoenicians,  and  Hebrews  is  supposed  to  have 
been  of  a  similar  character. 


198  MUSICOLOGY 

4.  Our  own  musical  system  can  be  traced  back  directly  to 
the  Greeks,  and  still  farther  back  through  the  early  migrations 
into  Greece,  to  the  Persians  and  Hindoos. 

5.  In  the  ancient  four-stringed  Greek  lyre,  the  outside 
strings  were  tuned  to  the  interval  of  a  4th,  and  the  middle 
strings  were  more  or  less  varied.  The  lyre  thus  gave  a  series 
of  four  notes,  which  was  called  a  tetrachord,  and  which  has 
ever  since  remained  a  prominent  element  of  the  music  scale. 

6.  About  670  B.C.  Terpander  increased  the  number  of 
strings  to  seven,  making  two  tctrachords  with  the  middle 
string  common  to  both.  The  tuning  depended  entirely  upon 
the  ear;  there  was  yet  no  means  for  accurately  measuring  or 
recording  the  pitch  of  tones. 

7.  This  seems  to  be  the  history  in  brief  of  the  development 
of  the  scale  up  to  the  time  of  Pythagoras,  about  550  B.C., 
who  is  regarded  as  the  father  of  musical  science. 

8.  Pythagoras  discovered  the  relations  between  the  length 
of  strings  and  the  pitch  of  tones,  and  that  intervals  could  be 
given  definite  numerical  values.  He  found  that  the  simplest 
division  of  the  string  (into  two  equal  parts)  gave  a  tone  which 
his  ear  told  him  was  the  most  closely  related  to  the  funda- 
mental, thus  fixing  the  interval  of  the  octave.  He  next  found 
that  two-thirds  of  the  string  gave  a  tone  which  formed  a 
natural  subdivision  of  the  octave,  and  also  that  three-fourths 
of  the  string  gave  another  natural  subdivision  of  the  octave. 
He  no  doubt  observed  the  relationship  of  these  two  intervals 
(now  called  perfect  5th  and  perfect  4th)  to  each  other; 
that  either  reckoned  upward  and  the  other  downward  gave 
the  interval  of  an  octave,  and  that  the  octave  therefore  was 
equal  to  their  sum.  Thus  these  three  intervals  were  estab- 
lished as  the  most  important  intervals  in  music. 

9.  The  tones  of  the  scale  thus  fixed  (taking  the  octave  on 

C)    are    C F    G C,    leaving    a    perfect 

4th  between  C  and  F  and   between  G  and   C  to  be  filled   up. 
Since  the  5th  and   4th   are   natural   intervals,    their  difference 


APPENDIX 


199 


(now  called  a  tone)  is  also  a  natural  interval,  as  it  occurs 
naturally  between  F  and  G.  This  is  taken  as  the  most  con- 
venient interval  to  subdivide  the  perfect  4ths.*  This  led 
to  an  unequal  division,  as  the  perfect  4th  contained  two 
tones,  and  a  semitone  (nearly)  over,  and  admitted  of  three 
arrangements,  namely,  the  semitone  may  be  below,  in  the 
middle,  or  above,  all  of  which  were  used.  Thus  the  octave  was 
made  up  of  two  tetrachords  with  a  separating  tone  in  the  middle. 

10.  Having  established  the  scale  of  the  octave,  it  was  easy 
to  extend  the  scale  by  adding  octaves  of  the  notes  already 
fixed.      In  this  way  the  scale  was  extended  to  two  octaves. 

1 1.  The  scale  thus  established  has  remained  practically  un- 
changed for  more  than  2000  years  to  the  present  time.  How- 
ever, a  slight  correction  was  found  necessary.  The  difference 
between  the  perfect  4th  and  perfect  5th  was  a  major  tone 
(ratio  I) ;  using  this  as  a  measure,  the  Pythagorean  scale 
would  be  as  follows : 

T  9814  S27248/, 

CDEFGABC 

The  two  major  tones  together  form  the  interval  of  a  Pythag- 
orean major  3d  having  a  ratio  |  X  |  =  f  j,  which  is  inhar- 
monious;  but  we  find  very  near  it  the  simple  harmonious 
ratio  |f=f,  which  is  called  the  true  major  3d.  The  differ- 
ence between  the  true  major  3d  (|)  and  the  major  tone  (I)  is 
J -f-  I  =  \'*,  which  is  called  the  minor  tone.  The  difference 
between  the  Pythagorean  and  the  true  major  3d  (f^  -^  ||)  is 
|i.  This  added  to  the  semitone  gives  |f-|  X  f i  =  t|.  The 
corrected  scale  then  is  as  follows : 

T  »  E  4  a  SlSn 

CDE-FGABC 

T  V  »  vioviov  *  V  i0v'9v  i«  —  o 
I    A   g  A  -j-AxsA  g  A  -5   Aft-  A  y?  —  2 

♦Another  mode  of  construction  is  by  perfect  sths  and  octaves.  Thus,  beginning  at  C  and 
taking  sths  upward,  the  tirst  will  give  G,  the  next  D;  dropping  down  an  octave,  the  next 
will  give  A,  the  next  E;  dropping  down  an  octave,  the  next  will  give  B;  then,  taking  a  5th 
downward  from  C,  we  get  F.  This  is  practically  the  same  as  alternately  taking  sths  up 
and  4ihs  down,  since  a  5th  up  and  an  octave  down  is  the  same  as  a  4th  down. 


200  MUSICOLOGY 

which    we    see    by    the    simpler     ratios     is   more    harmoni- 
ous. 

12.  The  major  3d  in  the  lower  tetrachord  could  be  corrected, 
either  by  shortening  the  interval  between  C  and  D  or  between 
D  and  E ;  but  D  is  an  important  note,  as  it  forms  a  perfect 
5th  to  the  G  below,  G  being  second  in  importance  to  the 
tonic  C ;  the  correction  is  therefore  made  by  shortening  the 
interval  between  D  and  E.  In  the  upper  tetrachord  the  cor- 
rection is  made  by  shortening  the  interval  between  G  and  A, 
thus  correcting  the  major  3d  between  F  and  A  as  well  as 
the  major  3d  between  G  and  B.  This  puts  the  I,  IV,  and 
V  triads  in  perfect  tune;  but  if  A  were  major,  the  IV  triad 
(F  A  C)  would  be  out  of  tune. 

13.  Observe  also  that  the  minor  3d  between  D  and  F 
(ratio  V  X  if  =  If)  is  not  a  true  minor  3d,  while  all  the 
other  minor  3ds  are  true  minor  3ds  (ratio  |  X  if  =  t)  !  also 
that  the  ratio  of  the  5th  between  D  and  A  is  VX  if  X|XV 
=  ij,  while  the  ratio  of  the  perfect  5th  is  |;  therefore  the  li 
triad  (D.  F.  A.)  is  still  not  in  perfect  tune — its  minor  3d  and 
its  5th  being  too  flat  by  the  difference  between  the  major  and 
minor  tones  (f -^-V  =  ll),  called  a  Comma,  or  a  little  more 
than  one-fifth  of  a  semitone  (log.  of  |J-  is  .00539;  ^^S-  of 
semitone  is  .02509  :  /^W  =  i  nearly).  This  is  a  necessary 
evil,  since  it  can  be  remedied  only  by  shortening  the  interval 
between  C  and  D,  instead  of  between  D  and  E,  which  would 
put  the  V  triad  out  of  tune. 

14.  The  correction  of  the  Pythagorean  major  3d  to  a  true 
major  3d  was  first  suggested  by  Didymus  about  the  beginning 
of  the  Christian  era,  but  was  not  fully  adopted  till  toward  the 
close  of  the  middle  ages. 

15.  These  distinctions  do  not  apply  to  the  equal  tempera- 
ment scale,  all  of  the  intervals  of  which,  except  the  octave, 
are  more  or  less  out  of  tune;  temperament  being  merely  an 
attempt  to  conform  the  scale  to  a  practical  simplicity  in  keyed 
instruments.      The  tempered   scale  has  come  gradually  into 


APPENDIX  201 

use  with  the  growth  in  popularity  and  prominence  of  keyed 
instruments. 

1 6.  The  Greeks  named  the  fifteen  tones  in  the  two  octaves 
by  different  Greek  words.  The  Romans  adopted  the  Greek 
scale  but  named  the  fifteen  tones  by  the  first  fifteen  Roman 
letters  from  A  to  P.  Gregory,  about  600  A.D.,  adopted  the 
system  of  repeating  the  same  letters  in  each  octave,  but  used 
capitals  in  the  first  octave,  small  letters  in  the  second  octave, 
and  double  small  letters  in  the  third  octave ;  which  distinc- 
tions were  finally  dropped. 

17.  Guido  of  Arezzo,  about  the  beginning  of  the  eleventh 
century,  discarded  the  tetrachord  method  of  grouping  the 
tones  and  arranged  the  twenty  notes  then  in  use  into  groups 
of  six,  called  hexachords ;  and  to  facilitate  the  singing  named 
the  six  notes  Ut,  Re,  Mi,  Fa,  Sol,  La,  which  were  suggested 
by  a  verse  of  a  hymn  to  St.  John,  in  which  they  occurred  in 
order  as  the  first  syllable  of  each  successive  line,  each  line  be- 
ginning a  note  higher.  Ti  has  been  added  to  fill  out  the 
octave ;  and  Ut,  being  a  bad  syllable  to  sing,  has  been 
changed  to  Do.  Guido  also  introduced  the  system  of  nota- 
ting  the  scale  on  lines  and  spaces. 

18.  Franco  of  Cologne,  about  the  middle  of  the  twelfth 
century,  is  credited  with  introducing  the  system  of  represent- 
ing the  relative  time  value  of  notes  by  their  shape.  The  sys- 
tem of  dividing  the  music  into  measures  by  drawing  bars 
across  the  staff  can  be  traced  to  about  the  year  1574. 

19.  The  characters  ff,  b,  and  <  originated  during  the  eleventh 
and  twelfth  centuries.  The  y  and  the  S  came  from  different 
forms  of  the  letter  J3  and  were  used  originally  to  show  the 
position  of  that  letter  only- — 1?  showing  the  lov/er,  and  t|  the 
higher  position.      The  H  originally  was  a  St.  Andrew's  cross. 


202  MUSICOLOGY 

ANCIENT  GREEK    MODES 

1.  Mode  refers  to  the  octave  form,  or  mode  of  arrange- 
ment, of  the  steps  and  half-steps  in  the  octave.  On  page 
198  :  9,  the  octave  was  shown  as  made  up  of  two  tetrachords 
with  an  added  (separating)  tone  between  ;  the  added  tone 
was  sometimes  placed  below  or  above,  instead  of  between. 

2.  The  tetrachord  may  be  arranged  in  three  principal  ways, 
as  follows  (letting  —  represent  the  step  and  ^  the  half-step) : 
—  ^  ^  (Lydian  tetrachord) ;  _  ^  —  (Phrygian  tetra- 
chord) ;  ^  _  _  (Dorian  tetrachord).  The  names  refer  to 
three  people  or  provinces  of  Greece,  from  whom  they  are  sup- 
posed to  have  originated. 

3.  The  modes  or  octave  forms  were  termed  Lydian,  Phryg- 
ian, or  Dorian,  according  to  the  kind  of  tetrachords  involved. 
If  the  added  tone  was  below,  the  term  Hypo  was  prefixed; 
if  the  added  tone  was  above,  the  term  Hyper  was  prefixed. 
There  was,  however,  some  variation  in  the  names,  as  shown 
below. 

4.  The  Earlier  Greek  Modes.  These  were  seven  in  num- 
ber, as  follows : 


1.  Lydian  C—^D—  £  ^  F-   t  —  J\  —B  -    C 


2.  Phrygian  J)  -  E  ^  T -    C  -  71  -  B  ^   C  -  J) 


3.  Dorian  JS -- T -   C  -  Jf  -S'^^^-D-S 

4.  Hypo-Lydian  (Syntono-Lydian)  T —   C  ~  ^  ~ -B  ^  Q —  J)  —  E  ^  T 

5.  Hypo-Phryjfian  (Ionian  or  lastian)       C—  J\  —  S  "^  C —  Jf  —  E  ^  F —  w 


6.  Hypo-Dorian  (.^^olian  or  Locrian)      Ji —  ^^   C — J)  —  £1  '~  F —  G — Ji 


^    j  Hyper-Dorian  ^  j  ^^^l^-^n/  -  F-  C  -  A  -  B 

\  Mixo-Lydian    S  [  £^   C-J)JJ^f—  C'^^W^  B^  {€) 

(See  also  p.  54,  Fig.  13.) 

5.   We  see  that  all  these  modes,  if  starting  on  the  proper 
letter  as  shown  above,  exactly  fit   into  the  scale  without    in- 


1234567 

f#        1 

4i        .  1  1  i  1  - 

APPENDIX  203 

volving  accidentals  (corresponding  to  our  keys  on  the  same 

letters,   with   the   signatures  omitted).      In   this  sense,    each 

mode  represents  a  certain  section 

of  the  scale,  as  indicated  in  Fig. 

49.      It  should   not    be  inferred, 

however,  that  they  were  used  only 

in  this  sense.     They  may  each  be 

taken  at  any  pitch,  which  would  fig.  49. 

thus    involve    accidentals   to    express    them  in    our  notation 

(remembering  that  ^'s,  b's,  and  also  the  staff  and  roman  letters 

were  not  used   by  the  Greeks).      In    either  sense,  the  modes 

represent  different  octave  forms. 

6.  The  Greek  music  (as  all  ancient  music)  was  monophonic. 
The  melodies  were  at  first  confined  to  the  compass  of  the 
tetrachord ;  later,  two  tetrachords  were  combined  and  the 
scale  thus  extended  to  the  octave ;  and  still  later,  the 
scale  was  extended  to  two  octaves.  In  the  meantime,  the 
Hypo-Dorian  (corresponding  to  our  old  minor  mode,  see  p. 
53:  I,  2)  became  the  mode  in  most  common  use,  and  thus 
acquired  prominence  over  the  others  in  a  manner  analogous 
to  our  major  mode, 

7.  The  Later  Greek  Modes.  About  the  fourth  century 
B.C.,  at  the  time  of  Aristoxenus,  the  scale  had  been  ex- 
tended to  two  octaves,  and  there  were  thirteen  modes  in  use, 
which  were  afterward  increased  to  fifteen,  as  shown  on  p. 
204. 

8.  These  might  also  be  represented  on  the  staff,  similarly 
as  in  Fig.  49,  by  placing  before  each  a  signature  correspond- 
ing to  the  number  of  sharps  or  flats  in  each. 

9.  It  will  be  seen  that  the  modes  are  named  according  to 
the  character  of  the  octave  between  the  dotted  lines  in  which 
all  the  modes  stand  parallel.  It  will  be  seen  also  that  as  be- 
tween the  extremes  of  each  they  are  all  the  same,  differing 
only  in  pitch,  and  are  merely  transpositions  of  the  Hypo- 
Dorian.      In  this  sense,  therefore,  they  are  merely  transposing 


204 


MUSICOLOGY 


PQ 


Ft;      Pl^^ 


) 

I 


) 


I 

I 


) 

ft; 


I     J       I 


"^ 


S   '=^ 


I 

>  ^)>     ) 

cq       ft:,      cq 


-— -V-vS«J 


f*^     fs     fs     K 
I     J       I     J- 


i<,  to  t<i 

h'            '  I 

c:^  K-,  R^ 

I         I  ) 


ft)      Pq      cq 
) 


(<1      fe^      kq 


I     { 


i 
) 


(5^1     r:^     K, 


) 

I 


'^       ^       ^       'O 

^      !       I       I 

nq      ft^      fiC,      0:^      Cq 


K 


I 


>         I 

) 

•« 

1 


•i 


I 
I 

I 


,1 

■      I 


<0 


<0 

<!  >  < 


"^ 


) 


J  < 


I     { 


.RjJ.-B^ 


}  I 


J 


I 


) 

J 

J 


-I 


-Rt; p^..^g...e^._f5^- 


) 


> 


4 


<ij       <0        "^3  'O 

k,      K       k,  k,  k  k, 

\     J        ^  \  J  \ 

^      kj      kj  ki  ^  k) 

1  )  )  I 


I       I 

>         I 


a 


►2     o 


U3 


^ 


S      a 


cij        q^      o:, 


*T3 

O.C 

c 

c 

$.2 

c 

.2 

2 

•^     OJL 

rt 

-3 

'bb 

"s  S* 

"o 

^ 

^ 

6 

O    o 
1— '    a. 

APPENDIX  205 

scales  (corresponding  to  our  fifteen  old-minor  keys).  How- 
ever, they  may  have  been  used  largely  in  a  parallel  sense  as 
distinct  octave  forms. 

10.  There  has  been  much  dispute  as  to  their  real  character, 
some  holding  the  theory  of  transposing  scales,  and  others  the 
theory  of  distinct  octave  forms. 

11.  The  Early  Church  Modes.  In  the  fourth  century 
A.D.,  Bishop  Ambrose  of  Milan,  in  view  of  the  existing  con- 
fusion, established  four  scales  for  church  use,  corresponding 
to  the  original  Greek  Phrygian,  Dorian,  Hypo-Lydian,  and 
Hypo-Phrygian,  and  established  also  their  character  as  dis- 
tinct modes,  or  octave  forms,  by  fixing  the  lowest  note  of 
each  as  the  tonic.  A  little  later  Pope  Gregory  added  to 
these  four  others  derived  from  them  by  ranging  the  scale 
from  a  4th  below  the  tonic  to  a  5th  above.  These  were 
called  P/a^d/,  while  the  Ambrosian  ones  were  caWedAut/u'^iftc. 

12.  In  time,  much  confusion  again  existed,  and  in  1547 
Glareanus  proposed  six  authentic  and  six  plagal  modes,  mak- 
ing twelve  in  all,  which  he  called  the  Dodecachordon.  Of 
these,  the  ten  which  remained  are  given  in  the  following  out- 
line : 


Church  name 


Name  given  by  Original 

Glareanus  Greek  name 


fist  mode       (Authentic)  Dorian  Phrygian 

2d       "  (Plagal  of  1st)  Hypo-Dorian 

3d       "  (Authentic)  Phrygian  Dorian 

4th     "  (Plagal  of  3d)  Hypo-Phrygian 

•g  1  5ih     "  (Authentic)  Lydian  Hypo-Lydian 

S    I  6th     "  (Plagal  of  5th)  Hypo-Lydian 

U    I  7th     "  (Authentic)  Mixo-Lydian  Hypo-Phrygian 

[sth     "  (Plagal  of  7th)  Hypo-Mixo-Lydian 

Secular  ^  Our  Major  Mode  Ionian  Lydian 

Modes    "(  Our  (old)  Minor  Mode   ^olian  Hypo-Dorian 

Observe   that   by   some  confusion    Glareanus  misapplied   the 
original  Greek  names. 

13.   The  eight  modes  as  established  by  Ambrose  and  Greg- 
ory are  known  as  the   Early  Church  modes.     The  other  two, 


2o6 


MUSICOLOGY 


corresponding  to  our  major  and  minor  modes,  were  used  only 
in  secular  music  ;  but  these  last  are  the  only  ones  that  have 
survived  through  the  development  of  polyphonic  music,  since 
they  are  capable  of  more  harmonic  combinations  than  the 
others. 


TABLE    OF  COMMON    MUSICAL   INTERVALS 


Intervals 

Ratio 

Semitone 
value 

Intervals 

Ratio 

Semitone 
value 

(  Perfect 

\ 

1                 r  Minor 

1* 

8.14 

Unison  •;   .              \ 
(Aug.      -^ 

minor 
semitone 

S6    * 

.70 

6th  ■  ^^J*^^ 

s  * 
s 

8.84 

Minor   - 

major 
semitone 

1  6   * 

1. 12 

[Aug. 

("Dim. 
7th  1  Minor 

8SS 

9.76 
9.26 

2d     Major  •] 

minor  tone 
major  tone 

V  * 

1     * 

1.82 
2.04 

¥ 

9.96 

[Aug, 

II 

2.74 

[  Major 

1 5* 

"8" 

10.S8 

(  Dim. 
3d  \  Minor 
(  Major 

m 
1* 

1* 

2.24 
3.16 
3.S6 

/->  .         S  Dim. 
^^'^-"^  \  Perfect 

1 

11.29 
12.00 

(  Dim. 
4th  "I  Perfect 
(Aug. 

4.2S 
4.98 
5.90 

■  Comma 

I  Pythagorean  Comma 

IIHIi 

.22 
.24 

(  Dim. 

5th  ^  Perfect 

(Aug. 

3    * 

6.10 
7.02 

Enharmonic  Diesis 

1  S8 

.42 

If 

7.72 

Schisma 

»S80B 

.02 

1.  The  derivations  of  the  ratios  marked  with  a  *  are  given 
on  pp.  157:2;  159:11;  160:15.  From  these  the  others  are 
easily  found. 

2.  If  we  subtract  (  -^  )  the  major  semitone  E  to  F  from 
the  major  3d  E  to  G||,  we  get  the  aug.  2d  F  to  G^  (f  -^  j|  = 
-g^).      The  divi.  7th  is  tlTe  inversion  of  this  (i--^1t=-'t/)' 

3.  If  we  add  (  X  )  two  major  semitones  we  get  the  dim. 
3d  (If  X  11=  III).      The   aug.    6th   is   the  inversion   of  this 

/8   .i.  256    —   885N 


APPENDIX  207 

4.  If  we  add  two  major  3ds  we  get  the  aug.  5th  (|  X  |=f|). 
The  dim.  4th  is  the  inversion  of  this  (f  -r-  y|=  it)- 

5.  The  aug.  4th  (tritone)  from  F  to  B  equals  the  major 
tone  F  to  G  plus  the  major  3d  G  to  B  (|  X  f  =  f|).  The 
dim.  5th  is  the  inversion  of  this  (t  ^~  If  =  ID- 

6.  The  minor  7th  is  the  inversion  of  the  major  2d  (f  -j- 1 
=  V")-     The   major   7th   is   the  inversion   of    the   minor   2d 

7.  The   dim.  octave  is   the   inversion   of  the   aug.    unison 

/2    _L_    85    . iH\ 

\J      -ST  —  Sj)- 

8.  The  Coninia  is  the  difference  between  the  major  and 
minor  tones  (f  -^  V-  =  |f ).  This  comma  is  always  meant 
unless  otherwise  designated. 

9.  The  Pythagorean  Comma  is  found  by  taking  twelve  5ths 
ap  and  seven  octaves  down,  thus: 

Sv/SvSv     3v3v3v3v3v-'5v3V-''V3v1     VlV^V 
^   A    3    A  ^    A     3    A    2^   A    3    A   -g^  /\    g-  /\   ^"   A    g'   /N    3    /\   -g-  /^    2    /S    2   /N    2    /\ 

iXiXiXi  =  i^=  fliHi-      It  is  also  equal  to  the  sum  of 
the  comma  and  schisma  (.22  +  .02  =  .24). 

10.  The  Enharmonic  Diesis  is  the  difference  between  the 
major  and  minor  semitones  (If  ^It^^tH)- 

1 1.  The  Schisma  is  found  by  taking  eight  5ths  and  a  major 
3d  up  and  five  octaves  down,  thus: 

ix|x|x|x|x|-x|x|xfxixixixix  i=|fm. 

It  is  also  equal  to  yV  of  a  Pythagorean  Comma  (.24-7-    12  = 
.02). 

TEMPERAMENT 

I.  Temperament  is  an  attempt  to  conform  the  scale  of  na- 
ture to  the  keyboard.  From  the  earliest  invention  of  the 
keyboard  it  has  contained,  in  general  practice,  twelve  keys  or 
notes  to  the  octave.  A  few  attempts  have  been  made  to 
make  the  keyboard  more  nearly  conform  to  the  scale  of  na- 
ture by  increasing  the  number  of  keys;  but  such  instruments 
were  too  complicated  and  never  came  into  general  use,  so 
that  twelve  remains  as  the  most  practical  number.      Hence 


208  MUSICOLOGY 

the  object  of  the  twelve-note  systems  of  temperament  is  to 
limit  the  scale  to  twelve  notes  to  the  octave,  so  tuned  as  to 
give  the  best  practical  results. 

2.  There  are  only  two  systems  of  temperament  that  have 
been  used  to  any  extent :  the  Equal  Temperament,  which  is 
the  one  now  in  most  general  use ;  and  the  Mean-  Totie  Tem- 
perament, which  was  the  one  formerly  in  most  general  use, 
and  is  still  at  least  of  historical  interest. 

3.  Equal  Temperament,  however,  is  the  older.  It  was 
very  early  observed  that  twelve  5ths  minus  seven  octaves 
equaled  the  Pythagorean  comma  (about  |  of  a  semitone), 
and  Aristoxcnus  (fourth  century  B.C.)  is  said  to  have  advo- 
cated the  idea  of  distributing  this  difference  among  the  twelve 
5ths  (the  5ths  thus  tempered  are  called  equal  5ths).  This  is 
the  basis  of  equal  temperament;  for,  taking  twelve  equal  5ths 
up,  thus,  C,  G,  D,  A,  E,  B,  F  #.  C  #,  GJt,  D^^,  A  t,  E  #  or  F,  B  jj 
or  C,  or  twelve  equal  5ths  down,  thus,  C,  F,  B1^,  Eb,  AI',  T)^, 
GK  Ci'  or  B,  Fb  or  E,  A,  D,  G,  C,  we  get  the  entire  chro- 
matic scale.  In  either  case,  tAvelve  5ths  takes  us  once  around 
the  key-circle  (which  is  7  octaves)  to  the  starting-point ;  but 
if  the  5ths  were  true,  we  would  have  landed  a  Pythagorean 
comma  past  the  starting-point.  If  we  had  dropped  down  an 
octave  whenever  we  exceeded  that  interval,  or  had  alternately 
taken  5ths  up  and  4ths  down  (two  successive  4ths  down  from 
B  to  C^),  we  would  have  gotten  the  chromatic  scale  direct. 

4.  The  great  advantage  of  equal  temperament  is  that  all  the 
keys  are  equally  in  tune  and  may  all  be  used.  The  principal 
objection  is  the  sharp  major  3ds  which  result  from  this  tem- 
perament ;  to  remedy  which,  the  Mean-Tone  Temperament 
w^as  invented, 

5.  Mean-Tone  Temperament  can  be  traced  back  to  Zarlino 
and  Salinas,  two  Italian  authors  of  the  sixteenth  century,  and 
in  1700  was  in  general  use  and  continued  in  general  use  till 
about  1840,  when  the  practice  in  tuning  keyboard  instru- 
ments began  to  return  to  Equal  Temperament. 


APPENDIX  209 

6.  Mean-Tone  Temperament  is  based  on  the  principle  that 
four  5ths  up  and  two  octaves  and  a  major  3d  down  (I  Xf  X  2  X 
|XiXiX|  =  |o-)  give  the  comma;  and  hence,  if  in  tuning 
by  5ths  up  and  octaves  down  we  use  5ths  diminished  by  i  of  a 
comma  (called  mean-tone  5ths),  we  would  get  true  major 
3ds.  The  difference  between  the  major  tone  (f)  and  the 
minor  tone  (V)  also  equals  the  comma  (§H-V'  =  |i-) ;  hence, 
if  we  flat  the  major  tone  or  sharp  the  minor  tone  by  |  of  a 
comma,  we  would  get  the  mean-tone,  or  tone  midway  be- 
tween. Now  if  we  begin  at  C  and  tune  by  mean-tone  5ths, 
the  first  5th  up  will  give  G,  which  will  be  i  of  a  comma  flat; 
taking  a  5th  downward  from  C  above  will  give  F,  which  will 
be  :f  of  a  comma  sharp :  therefore  the  interval  between  F 
and  G  is  contracted  |-  of  a  comma  and  becomes  a  mean-tone. 
Now  a  5th  up  and  an  octave  down  is  the  same  as  a  4th  (in- 
verted 5th)  down,  and  tuning  an  octave  down  after  every 
two  5ths  up  is  the  same  as  alternately  taking  5ths  up  and  4ths 
down  ;  and  therefore  every  whole  tone  thus  fixed  is  neces- 
sarily equal  to  the  difference  between  the  4th  and  5th,  since 
no  other  intervals  are  used.  Therefore  tuning  by  mean-tone 
5ths  necessarily  makes  all  the  tones  mean-tones;  from  which 
fact  the  temperament  takes  its  name. 

7.  The  comma  (|^)  is  about  1  of  a  semitone,  and  -\  of  a 
comma  would  be  about  aV  of  ^  semitone,  and  therefore  the 
mean-tone  5th  is  about  ^V  of  ^  semitone  flat ;  while  the  equal 
5th  of  the  equal  temperament  is  about  --V  of  a  semitone  flat. 
However,  the  error  is  still  quite  imperceptible  in  the  chord. 

8.  The  minor  3d  is  also  gV  of  a  semitone  flat  (being  the 
difference  between  the  true  major  3d  and  the  mean-tone  5th), 
while  in  the  equal  temperament  it  was  J  of  a  semitone  flat, 
and  is  therefore  much  improved. 

9.  In  the  following  table  we  have  mean-tone  tempera- 
ment, true  intonation,  and  equal  temperament  compared. 
The  second  and  third  columns  correspond  to  the  table  on 
p.  166. 


2IO 


MUSICOLOGY 


Mean-Tone 

True 

Equal 

Temperament 

Intonation 

Temperament 

Semitones 

Semitones 

Semitones 

c 

1 2.  GO 

12.00 

12.00 

R 

10.83 

10.8S 

11.00 

A 

8. 89 

8.84 

9.00 

G 

6.97 

7.02 

7.00 

F 

5-03 

4.98 

5.00 

E 

3-86 

3-86 

4.00 

D 

1-93 

2.04 

2.00 

C 

0. 

0. 

0. 

10.  To  find  the  values  in  the  first  column,  first  find  the 
value  of  the  comma,  thus:    3.86  —  2.04  =  1.82,  equals  minor 

tone  between  D  and 
E;  2,04  —  1.82  = 
.22,  equals  difference 
between  major  and 
minor  tones,  equals 
comma ;  .22  ^-  4  = 
.05  =  ;f  of  comma, 
equals  amount  of 
temperament  of  the 
mean-tone  5th. 

1 1.  7.02  —  .05  =  6.97  =  G ;  12.00  —  6.97  =  5.03  =  F ; 
we  may  take  E  directly  from  the  middle  column,  as  it  is  a 
true  major  3d;  3.86  ^  2  =  1.93  =  D,  or  mean-tone ;  5.03 + 
3.86  =8.89=  A;  6.97  +  3.86=10.83=6.  Or  by  taking 
5ths  up  and  4ths  down,  thus,  G  —  F=D,  D  +  G  =  A,  A  — 
F  =  E,  E  -j-  G  =  B ;  or  by  taking  5ths  up  and  octaves  down 
(whenever  the  octave  is  exceeded) ;  or  we  might  use  still  other 
combinations. 

12.  In  the  preceding  table  we  have  the  natural  scale  of  C, 
corresponding  to  the  white  notes  of  the  keyboard.  As  there 
are  only  five  black  notes  in  each  octave  of  the  keyboard  we 
can  extend  the  temperament  to  only  five  other  keys,  since  in 
mean-tone  temperament  the  sharps  and  flats  do  not  coincide 
as  in  equal  temperament.  We  might  take  the  first  five  sharp 
keys  or  the  first  five  flat  keys;  but  the  first  three  sharp  keys 
(G,  D,  and  A)  and  the  first  two  flat  keys  (F  and  Bl?)  are  the 
most  used. 

13.  The  first  three  sharps  are  FH,  Cit,  and  G#;  the  first  two 
flats  are  Bi?  and  Ek  To  tune  these  we  must  make  F|^  a  true 
major  3d  to  D  (1.93  +  3.86  =:  5.79) ;  C|?,  a  true  major  3d  to 
A  (8.89+ 3.86  —  12.00  =  .75);  G  it,  a  true  major  3d  to  E 
(3.864-3.86  =  7,72);  bI',  atrue  major  3d  below  D  (12.00 
+  1-93  —  3-86  =  10.07);    Ei?,  a  true  major  3d  below  G(6.97 


APPENDIX  2  I  I 

—  3.86  =  3. 1 1).  These,  with  the  white  notes,  would  give  us 
all  the  notes  of  the  six  keys  C,  G,  D,  A,  F,  and  B  1^,  all  equally 
in  tune.  The  other  keys  are  more  or  less  out  of  tune,  accord- 
ing to  the  number  of  notes  missing-  In  the  key  of  E,  D^  is 
missing  and  we  would  have  to  use  E  "?  in  its  place,  which  differs 
by  the  enharmonic  diesis  (see  table,  p.  206) ;  this  is  what  is 
called  a  wolf  (comparing  the  effect  to  the  howling  of  a  wolf). 
Similarly  in  the  key  of  E  1^,  we  would  have  to  use  GJI  for  A  1?. 
The  other  keys  contain  more  wolves. 

14.  In  some  old  organs  the  black  keys  between  D  and  E 
and  between  G  and  A  were  divided  and  tuned  to  give  both 
the  sharp  and  flat  notes,  thus  bringing  the  keys  of  E  and  E  ^ 
into  tune. 

15.  The  merit  of  the  Mean-Tone  Temperament  is  the 
greater  harmoniousness  of  the  most  useful  keys,  and  its  de- 
merit is  in  its  limitation  to  those  keys.  On  the  other  hand, 
equal  temperament  has  the  freedom  of  all  the  keys ;  for 
which  reason,  the  balances  finally  turned  in  favor  of  equal 
temperament. 

TUNING 

1.  Tuning  in  any  temperament  is  done  as  far  as  possible 
by  5ths,  4ths,  and  octaves,  till  all  the  notes  in  the  tuning 
octave  (usually  near  the  middle  of  the  keyboard)  are  tuned. 
These  are  called  the  bearings,  from  which  the  other  notes 
above  and  below  are  tuned  by  octaves, 

2.  Only  the  5ths  and  4ths  are  required  to  beat,  as  the 
octave  is  always  a  true  interval  and  should  not  beat.  The  5ths 
and  4ths  are  first  tuned  true,  then  the  5ths  contracted  and  the 
4ths  expanded   till  the  required    number  of  beats   are  heard. 

3.  The  usual  method  of  tuning  the  bearings  consists  in  al- 
ternately taking  5ths  up  and  4ths  down.  If  we  take  the 
octave  above  middle  C  as  the  tuning  octave,  the  notes  would 
be  tuned  in  the  following  order  (C  being  first  tuned  by  a 
standard  tuning-fork  or  other  standard). 

,.  A 


2  12  MUSICOLOGY 

EQUAL  TEMPERAMENT 

1.2      1.3       1.4       1.5     1.6       1.7      .9         .g        i.o       I.I       I.I      1.2 

C     Cf     D     D«     E     F     F«     G     GJf     A     A«     B     C 

72       94      II      61        83       10      5     12 
Fiu.  50. 

4.  The  figures  below  the  letters  show  the  order  in  which 
the  notes  are  tuned:  first,  taking  a  5th  up  to  G;  second,  a 
4th  down  to  D;  third,  a  5th  up  to  A;  etc.  The  figures 
above  the  letters  are  the  number  of  beats  per  second  in  each 
5th  up  or  4th  down  (the  number  varying  with  pitch).  First, 
calculate  the  number  of  beats  in  the  5th  and  4th  at  a  certain 
pitch,  then,  to  find  the  number  at  any  other  pitch,  multiply  or 
divide  by  the  ratio  of  the  interval  at  which  the  5th  or  4th  is 
taken  above  or  below  the  pitch  at  which  the  beats  are  already 
found. 

5.  Let  i5  =  number  of  beats;  iV=  number  of  vibrations 
in  the  lower  note;   71/=  number  of  vibrations  in  the  upper 

note;   —  =  the  ratio  of  the  true  interval  in  its  lowest  terms: 

then  the  algebraical  formulas  for  finding  the  number  of  beats 
would  be  (i)  ^=  Nni — Mn  if  the  interval  is  flat  (less  than 
the  true  interval),  or,  {2)  B  =^  Mn — Nni  if  the  interval  is 
sharp  (greater  than  the  true  interval). 

6.  It  was  shown  on  p.  167:  2  that  the  blending  harmonics 
were  in  exact  accord  with  the  ratio  of  the  interval ;  thus,  in 
the  5ths  (ratio  |)  every  second  harmonic  of  the  upper  note 
and  every  third  harmonic  of  the  lower  note  blend,  therefore 
multiplying  the  lower  note  by  the  upper  figure  of  the  ratio 
or  the  upper  note  by  the  lower  figure  would  give  the  lowest 
blending  harmonics,  and  similarly  for  any  interval ;  so  that 
the  number  of  vibrations  in  the  lowest  blending  harmonics  of 
two  notes  may  be  found  by  multiplying  the  vibrations  of  the 
lower  note  by  the  upper  figure  of  the  ratio  or  the  vibrations 
of  the  upper  note  by  the  lower  figure  of  the  ratio.  If  the 
interval  is  true,  these  products  will  be  equal  (.Vw  ^^  Mii)  and 


APPENDIX  213 

the  lowest  blending  overtones  will  be  in  unison;  but  if  the 
interval  is  not  true,  the  products  will  differ,  and  the  lowest 
blending  overtones  will  beat  by  not  being  in  perfect  unison. 

7.  It  was  shown  also  (p.  171  :  10)  that  the  difTerence  between 
the  lowest  blending  overtones  determined  the  number  of  im- 
perfect consonance  beats  ;  therefore,  Nm —  Mn  or  Mn  —  Nm 
(according  as  the  interval  is  flat  or  sharp)  equals  number  of 
beats. 

8.  The  amount  of  temperament  of  the  equal  5ths  is  jV  of 
the  Pythagorean  comma  (see  p.  208 :  3) ;  this  is  called  the 
Schisma,  the  ratio  of  which  is  ||41l  (see  table,  p.  206). 

9.  If  middle  C  =  256  vibrations,  G  =  256  X  |  =  384. 
Flat  G  by  the  schisma,  thus,  384  -^  Iff f|  =  383.56 +. 
Therefore  N  =  256,  M=  383.56,  ;;/  =  3,  ;;  =  2.  Substitu- 
ting these  values  in  formula  ( I ),  we  get  ^=  .88.  If  middle  C 
=  270  vibrations  (the  highest  standard),  we  would  get,  by 
figuring  from  this  standard,  B  =  .92.  Therefore  .9  (which 
is  the  mean)  may  be  taken  as  the  number  of  beats  in  the  5th 
above  middle  C  for  all  standards  of  pitch. 

10.  If  middle  C  =  256,  F  =  256  X  |  =  341.33.  Sharp  F 
by  the  schisma,  thus,  341.33  X  ||4f|  =  341 .72.  Therefore 
A^=2  56,  i/=  341.72,  ;;/ =  4,  ;/ =  3  ;  and  from  formula  (i) 
we  would  get  ^=1.16;  or  if  middle  €  =  270,  wt  would 
get  B  =^  1.23.  Therefore,  1.2  may  be  taken  as  the  num- 
ber of  beats  in  the  4th  above  middle  C  for  all  standards 
of  pitch. 

11.  From  these  values  we  may  obtain  the  number  of  beats 
at  any  pitch  ;  thus,  the  4th  from  G  down  to  D  is  a  major  2d 
higher  than  the  4th  from  C  to  F  (1.2  X -|  —  1.3),  which 
gives  1.3  for  D.  The  5th  from  D  up  to  A  is  a  major  2d 
higher  than  the  5th  from  C  to  G  (.9  X  |  =  i.o),  which  gives 
i.o  for  A,  etc. 


2  14  MUSICOLOGY 


MEAN-TONE  TEMPERAMENT 

3-3        3-6      30         41     3-3     4-<5        2.5     2.5        2.7     4.4       S-" 

C     C«      D     £!>     E     F     F«     G     G5     A     Bb     B     C 

72  4  6183  5 

3  I  2 

Fig.  51. 

12.  In  Mean-Tone  Temperament,  tune  the  notes  in  the  same 
order  as  before  up  to  Gjlf,  then  begin  again  at  C  and  tune  a 
4th  up  to  F,  then  another  4th  up  to  Bb,  then  a  5th  down  to 

13.  In  the  table  on  p.  206,  the  value  of  the  comma  is  .22  s. 
i  of  .22  s  =  .055  s,  which  is  the  amount  of  temperament  of  the 
mean-tone  5th.  Also  the  value  of  the  schismais  .02,  which  is 
the  amount  of  temperament  of  the  equal  5th  (.055  -^  .02  = 
24),  from  which  we  find  that  mean-tone  is  2|  times  equal  \.^xn- 
perament.  Therefore  the  simplest  method  of  finding  the 
beats  over  the  letters  in  Fig.  51  is  by  multiplying  the  number 
over  the  corresponding  letters  in  Fig.  50  by  2|,  except  F,  Bl?, 
and  El',  in  tuning  which  we  again  started  at  C;  but  we 
found  the  number  of  beats  for  C  to  F  in  equal  temperament 
was  1.2,  therefore  1.2  X  2f=  3.3  =  number  for  F  in  mean- 
tone;  F'  to  Bt?  is  a  4th  higher,  therefore  3.3  X  f  =  4.4  = 
number  for  Bl?;  B  1?  to  E  1?  is  a  minor  3d  higher  than  the  5th 
from  C  to  G,  therefore  2.5  X  |  =  3.0  =  number  for  E  b. 

14.  In  a  similar  manner,  we  might  calculate  the  beats  for 
any  other  tuning  octave  in  either  temperament. 

15.  It  is  necessary  to  count  the  number  of  beats  in  10 
seconds  to  remove  the  decimals;  3.3  beats  in  one  second 
equals  33  beats  in  10  seconds. 

CALCULATION   OF    PITCH    NUMBERS 

I.   From  formula  (i)  v^=  Nm  —  Mn  we  get  formulas  (3") 

^=  ^^  '"^"^  (4)  M=  ^1^.      From    formula    (2)    B  = 

Mn  -  Nm  we  get  formulas  (5)  iV  =  ^^"~  ^  and  (6)  M  =  '^'"\'^  ^' 


APPENDIX  2  I  5 

2.  Formula(3)  is  used  for  5ths  down  ;  (4)  for  5ths  up;  (5)  for 
4ths  down  ;   and  (6)  for  4ths  up. 

3.  In  taking  5ths  down  or  4ths  up  -5  is  +  because  the  tun- 
ing note  is  raised  to  contract  the  5th  or  expand  the  4th, 
while  in  taking  5ths  up  or  4ths  down  B  \s  —  because  the  tun- 
ing note  is  lowered  for  the  same  reason. 

4.  In  the  table  on  p.  216  the  number  of  vibrations  is  cal- 
culated for  each  note  of  the  octave  above  middle  C  for 
either  temperament.  The  number  of  the  formula  which 
applies  stands  opposite  each  section,  and  the  value  of  each 
letter  of  the  formula  is  indicated. 

5.  The  figures  over  the  letters  of  the  equal  tempered  scale 
are  practically  accurate  for  all  standards  of  pitch ;  but  owing 
to  the  greater  amount  of  temperament  of  the  mean-tone  5th, 
it  is  necessary  to  divide  the  range  between  the  extreme  stand- 
ards of  pitch  into  three  sections,  using  the  figures  over  the 
letters  of  the  mean-tone  scale  for  the  middle  section,  and 
adding  .1  to  each  for  the  higher  section,  and  subtracting  .1 
from  each  for  the  lower  section. 

6.  In  Mean-Tone  Temperament,  middle  C  ranges  in  the 
different  standards  from  252.7  (Handel's  pitch)  to  283.6 
(Durham's  pitch).  The  pitch  (264)  given  in  table  is  called 
Helmholtz's  pitch,  and  is  nearly  midway  between, 

7.  By  means  of  beats  we  may  also  find  the  number  of  vi- 
brations in  any  note,  without  the  known  vibrations  of  some 
other  note  from  which  to  figure. 

8 .  From  formula  ( i )  we  get B  =  [m  —  n  —  )  ^,  or  A^= r^ 

N 

D 

From  formula  (2)  we  would  get  iV= r,;  so  that  ^  is  +  or  — 

^   ^  m  —  n  izL 

N 

according  as  the  interval  is  flat  or  sharp. 

9.  Now  —  =  the  actual  ratio  of  the  interval  (being  the  vi- 
brations of  the  upper  note  divided  by  the  vibrations  of  the 
lower),  and  —  =  the  ratio  of  the  true  interval.      Let  it  be  re- 


Equal  Temperament 


1.2   1.3   1.4     1.5  1.6  1.7     .q    .9     i.o  l.i      I.I  12 

C  CJ  D   D$  E  F   FJG  G«  A  A«  B  C 

7        20        4     II    6       18        ^lo       5     12 


Mean-Tone  Temperament 


3-3  3-6  30  41  3-3  46    2.5  2.5  2.7  4.4    3.0 

CC5D   Eb  E   FF«GG«AB!JBC 

72  4        6183  5 

3  I  2 


fN  = 
I  m  = 
J 

1"  = 

L  n  = 
fM  = 


B  = 


256.0 
3 


768.0 


2)767.1 
.383-6 


II 50. 8 


=     4I114Q.5 


fN 
m 

B 

n 

= 

287.4 

3 

= 

862.2 

1.0 

2)861  2 

f  M 
n 

Lm 

f  N 
m 

= 

430.6 

3 

= 

I2QI.8 

IS 

Z 

4)1200.3 

322.6 

3 

967.8 


■M  = 


B 


=  B 


F| 


.  m  =    4)1084.8 


CS 


GJ 


=  D$ 


=  AJ 


fN  = 
;  ni  ■-= 
4    ■ 

B    = 

264.0 
3 

792.0 

2-S 

I  n    = 

2)789.5 

394.8 

3 

Mb  = 

1184.4 
3-6 

lm  = 

4)1180.8 

1  m  = 

295.2 
3 

I  n    = 

(M  = 

n    = 

885.6 
2.7 

2)882.9 
441.4  = 
3 

s  J 
|B   = 

1324.2 
4  I 

4)1320.1 

(-N  = 
m  = 
4 

B   = 

t  n    = 

'M  = 

n    = 

330-0 

3 

990.0 

3.0 

2)987.0 

4Q3.S 
3 

S  J 
»  B  = 
m  = 

.M  = 
n    = 

1480.5 
4.6 

4)1475.9 

36S.9 

3 

5 
|B  = 
lm  = 

I 106.7 
3-3 

4)1103.4 

m  = 

275-9 
3 

4  J 

B  = 

827.7 

2.5 

n    = 

2)825.2 

B 


.  m  =    4)1366.7 


FJ 


C« 


rN 
Jm 


412.6  =»  GJ 


264.0 


B  = 


1  B 

In 
(M 

J" 


1056.0 

3.3 

3) 

'0.S9.3 

353-1 

4 

1412.4 

4-4 

3) 

1416.8 

472-3 

2 

944.6 

30 

3 '947 -6 
3 '5-9 


Bb 


Bb 


APPENDIX  217 


quired  to  find  the  pitch,  or  number  of  vibrations,  in 


Tune  three  true  5ths  up  and  then  a  true  major  6th  down 
(f  X  f  X  I  X  I  —  li),  which  will  give  an  imperfect  octave 
above   the     given     note,    the  ratio  being   |^    instead    of   f ; 

AT 

hence  -  =  |4  and  ^  =  f-  Count  the  beats  made  by  this 
imperfect  octave  and  suppose  them  to  equal  32  in  10  seconds, 
or  3.2  per  second,  then  ^  =  3.2  ;  but  as  \\  is  greater  than  f 
the  interval  is  sharp,   so  that  B  is — 3.2.       Therefore    N  =^ 

zzj\  =  128.     (See  Poles,  Philosophy  of  Music,  p.  3  1 5 — Fourth 

4  0 

Edition.) 

CHARACTER   OF   THE   DIFFERENT    KEYS 

1.  There  is  a  noticeable  difference  in  the  effect  produced  by 
the  different  keys,  which  may  be  described  in  a  general  way 
as  follows : 

2.  The  key  of  C  major  expresses  earnestness,  resolution, 
and  decision.  Its  relative,  a  minor,  is  tender,  melancholy, 
and  plaintive. 

3.  G  major  (signature  i  sharp)  expresses  feeling  from  quiet 
and  calm  to  bright  and  joyous,  and  is  a  favorite  key.  Its 
relative,  e  minor,  is  mournful  and  persuasive. 

4.  D  major  (signature  2  sharps)  expresses  grandeur, 
triumph,  animated  feeling,  and  lofty  purpose.  Its  relative, 
b  minor,  is  very  melancholy  and  bewailing. 

5.  A  major  (signature  3  sharps)  expresses  sincerity,  hope, 
confidence,  love,  and  cheerfulness.  Its  relative,  f\  minor,  is 
dark,  mournfully  grand,  and  full  of  passion. 

6.  E  major  (signature  4  sharps)  is  the  most  brilliant  key, 
and  expresses  joy,  splendor,  and  magnificence.  Its  relative, 
c%  minor,  is  the  most  melancholy  key. 

.  7.  B,  or  C  I?  major  (signature  5  sharps  or  7  flats),  when 
loud,  expresses  pride  and  boldness;  when  soft,  expresses 
purity  and  clearness — not  a  favorite  key.      Its  relative,  g%  or 


2l8  MUSICCLOGY 

a?   minor,    has   a   peculiar,    sad,    tender  pathos,    suitable   for 
funeral  marches. 

8.  F  major  (signature  i  flat)  is  contemplative  and  ex- 
presses peace,  joy,  and  religious  sentiment.  Its  relative,  d 
minor,  expresses  anxiety,  solemnity,  and  grief. 

9.  B  b  major  (signature  2  flats)  is  bright  and  clear,  though 
not  specially  energetic  or  grand.  Its  relative,  g  minor,  is  sad, 
dreamy,  and  romantic. 

10.  Eb  major  (signature  3  flats)  is  full  and  soft  and  capa- 
ble of  a  great  variety  of  expression ;  its  general  character  is 
serious,  solemn,  firm,  and  dignified.  Its  relative,  c  minor,  is 
soft,  earnest,  longing,  and  passionate, 

11.  A  i^  major  (signature  4  flats)  is  sweet,  unassuming, 
delicate,  and  full  of  feeling.  Its  relative,  f  minor,  is  dismal 
and  gloomy. 

12.  D  i?  major  (signature  5  flats)  is  sublime,  deep,  and 
tragic.      Its  relative,  b'^  minor,  is  somber  and  mournful. 

13.  Gt?  major  (signature  6  flats)  is  very  soft  and  rich.  Its 
relative,  c^  minor,  is  full  of  melancholy  tenderness. 

14.  The  character,  however,  of  a  piece  of  music  does  not 
depend  wholly  on  the  key,  but  also  on  the  treatment  as  re- 
gards time,  rhythm,  etc.,  so  that  it  is  possible  in  this  way  to 
largely  counteract  the  character  of  the  key.  But  it  may  be 
observ^ed  that  the  character  of  the  same  piece  of  music  is  sen- 
sibly altered  by  a  change  of  key.  Therefore  the  character 
given  to  each  key  only  shows  the  general  tendency  of  each 
independently  of  other  considerations. 

15.  These  characters  cannot  be  due  either  to  a  difference 
in  the  intervals  or  to  temperament ;  for  all  keys,  if  equally  in 
tune,  regardless  of  temperament,  are  composed  of  the  same 
intervals  differing  only  in  pitch.  This  seems  to  indicate  that 
the  real  cause  is  the  difference  in  pitch.  This  view  is  sup- 
ported by  the  fact  that  pitch  affects  the  quality  of  tone  as 
regards  the  extent  and  prominence  of  the  harmonics. 


ABBREVIATIONS  USED  IN  MUSIC. 


The  followinof  Is  a  list  of  the  abbreviations  in  most  common 


use 


Accel. — Accelerando. 
Accom. — Accompaniment. 
Adg:. — Adagio. 
Ad  lib. — Ad  libitum. 
Affett.— Affettuoso. 
Affrett.— Affrettando. 
Ag. — Agitato. 
AH.— Allegro. 
Allgtt.— Allegretto. 
Air  Sva— Air  ottava. 
Al  seg. — Al  segno. 
And. — Andante. 
Anini. — Animato. 
Arp.— Arpeggio. 
A  tem. — A  tempo. 

Brill. — Brillante. 

C.  B.— Col  basso,  Contrab- 
basso. 

Cad. — Cadence. 

Cal. — Calando. 

Cant. — Canto. 

Cantab.— Cantabile. 

Ch. — Cboir  organ. 

Col.  ott.— Coir   ottava. 

Col.  vo.— Colla  voce. 

Con  esp.— Con  espressione. 

Cresc. — Crescendo. 

D.  C. — Da  capo. 
Decres. — Decrescendo. 
Diap.— Diapasons. 
Dim. — Diminuendo. 

P.  Mu8.— Doctor  of  Music. 

Del. — Dolce. 

D.  S.— Dal  Segno. 


Esp. — Espressivo. 

F.— Forte. 
Falset. — Falsetto. 
Ff. — Fortissimo. 
Fin. — Finale. 

F.  O.— Full  organ. 
Forz. — Forzando. 
F.p. — Forte- piano. 

G.  O. — Great  organ. 
Grand. — Grandioso. 

Intro. — Introduction 

L.— Left  hand. 
Leg. — Legato. 
liO. — Loco. 

Maest. — Maestoso. 
Magg. — Maggiore. 
Marc. — Marcato. 
Met. — Metronome. 
Mez. — Mezzo. 
Mf.  or  Mff. — Mezzo  forte. 
M.  M.  —  Maelzel's     metro- 
nome. 
Mod. — Moderate. 
M.  P. — Mezzo  piano. 
M.  V. — Mezza  voce. 

Obb.— Obbligato. 
Sva  or  8a— Ottava,  Octave. 
Sva  alta — Ottava  alta. 
Sva  bftH.— ottava  bassa. 


P. — Piano,  pousise. 

Ped.— Pedal. 

P.  F.  or  Pf.— Piano-forte. 

P.  f.— Piu  forte. 

Pianiss. — Pianissimo. 

Pizz. — Pizzicato. 

PP. — Pianissimo. 

4tte — Quartet. 

Ball. — Rallentando. 
Becit. — Recitative. 
B.H.— Right  Hand. 
Bitar. — Ritardando. 

S. — Segno,  senza,  sinistra, 

solo,  subito. 
Scherz.- Scherzando. 
Seg. — Segno,  segue. 
St.  or  Sfz.— Sforzando. 
Sos. — Sostenuto. 
Spir. — Spiritoso. 
Stacc. — Staccato. 

Tern. — Tempo. 

Tr.— Trillo. 

Trem. — Tremolando. 

U.  C— Una  corda. 
Unis. — Unison. 

V. — Verte,  voce,  vocl,  vol- 

ta,  volti. 
Var. — Variation. 
Viv. — Vivace. 


MUSICAL  DICTIONARY. 


Abbreviations— (F)    French;     (G)    German;     (I)  Italian;    (old  E)  Old  English; 
(S)  Spanish;    (L)    Latin;    (Gk)  Greek. 

A. 


A  ballata   (I)— In  the  ballad  style. 

Abbandono,  con  (I) — With  self-aban- 
donment,  passionately. 

A  battuta   (1) — In  strict  time. 

Abendlied    ((i) — An   evening  song. 

A  bene  placito   (I) — At  pleasure. 

Ab  Initio   (L) — From  the  beginning. 

A  rappelia  (I)  — (Ij  In  the  Church 
style;    (2)  Church  music,  duple  time. 

A  capriccio   (I) — At   will. 

Accelerando  or  Accelerato  (I) — Grad- 
ually increasing  the  pace. 

Acciaccatura    (I) — A   short   grace-note. 

Accoiade  (F) — A  brace,  uniting  several 
staves. 

Achromatic— Not  chromatic. 

A  cinque  (I) — In  live  parts. 

Adag:ietto  (I) — A  diminutive  of  Ada- 
gio ;    slower  than  Adagio. 

Adagio   (I) — Slowly. 

Adaffissimo  (I»— Very  slow  indeed. 

Additato  (I)— Fingered  ;  having  signs 
to  show  what  fingers  are  to  be  used. 

Addolorato  (I  I— In  an  afflicted  manner. 

Ad  libitunt    (D— At  will. 

Ad  piacitum   (L)— At  pleasure. 

AlTabile   (I)— In  a  pleasing  manner. 

Affannato  (I)— In  a  distressed  manner. 

AfTanno,  con   (I )— Mournfully. 

AfTannoso   (I)- Mournfully,  with  grief. 

Affetto,  con   (I)— With  affection. 

Aflfettuoso   (I )— Affectionately. 

Affezione,  con    (I)— With   tenderness. 

Affrettando   (I)— Hastening  the  time. 

A  fior  di  lalibra  (I }— Speaking  or  sing- 
ing very  softly  on  the  lips. 

Agevole  (I  (—With  facility,  lightness. 

Agilita,  con    (I)— With   sprightliness. 

Agilite  (F) — Lightness  and  freedom  in 
playing  or  singing. 

AKitamento   (I)— Restlessness. 

Agitato  ri)— An  agitated,  restless  style. 

A  ia  (F),  Al.  All',  Alia  (I)— Like,  in, 
at,  in  the  style  of. 

A  la  nieme  (F)— In  the  original  time. 

A  la  niesure  (F) — In  time. 

Alia  breve  (I)— A  direction  that  the 
notes  are  to  be  made  shorter. 

Air  «va  alta    (1)     The   octave   higher. 

All  8va  bassa   (I)— The  octave  lower. 

.AlleKramente    (I)     .loyfully,  cheerfully. 

.Vllegretto   (I)— Slower  than  Allegro. 

.Vllegrettino  (I)— Not  SO  fast  as  Alle- 
grctio. 

.Vllegrezza    (I)     .Toy,   rejoicing. 


Allegrissimo   (I) — Extremely  quick. 

Allegro   (I) — Literally,  Joyful.     Quick, 
lively.      In    music    it    is    sometimes 
qualified  as: 
Allegro  assai   (I) — A  quicker  motion 

than  simple  allegro. 
Allegro    con    brio    (I) — Quickly    and 

with  spirit. 
Allegro  con   fuoco   (I) — Rapidly   and 

with  fire. 
Allegro    con     moto     (I) — With    sus- 
tained joyfulness. 
Allegro     con     spirito      (I) — Joyfully 

and  with  spirit. 
Allegro  di  bravura  (I) — A  movement 

full  of  executive  difficulties. 
Allegro     furioso     (I) — Rapidly     and 

with  fury. 
Allegro   ma   nou   troppo    (I) — Lively, 

but  not  too  fast. 
Allegro     moderato      (I) — Moderately 

quick. 
Allegro  vivace  (I) — Lively  and  brisk. 

Aliemande  (F) — Alemain,  AUemaigne, 
Almain.     A  dance  in  duple  time. 

.\llentando    (I) — (Gradually    slackening 
the  time. 

All'  unisono  (I) — In  unison  or  octaves. 

.M  Negno   (I) — To  the  sign. 

Alt  (I) — The  notes  in  the  octave  begin- 
ning with  G  above  the  treble  stave 
iire  said  to  l)e  in  alt. 

.Mtieramente    lit      Troudly,   grandly. 

AltiNsimo  (It — The  highest.  The  notes 
in  the  octave  beginning  with  G  on 
the  fourth  leger  line  above  the  treble 
stave  are  s;ud  to  be  in  altissimo. 

Alto  clef— The  C  clef,  placed  upon  the 
third  line  of  the  stave. 

A  mezza  voce  (11  — (1)  With  half  the 
strength  of  the  voice;  (2)  The  qual- 
ity lictween  the  chest  and  head 
v(")ice;  (.'<)  The  subdued  tone  of 
instruments. 

Aniore,  con   (H — With  love,  affection. 

Andante  (1) — Literally,  Walking.  Slow, 
graceful,   distinct   and   peaceful. 

cantabile.  Slow,  and  in  a  sing- 
ing style. 

con  moto.     Faster  than   Andante, 

and  with  animation. 

Andantino  (I) — Slower  than  Andante. 

Angenebm    (G) — Pleasing,   agreeable. 

Anglaise  (F),  Anglieo  (I)— The  Eng- 
lish country  dance. 


MUSICAL   DICTIONARY. 


221 


Aniiuato  (I) — Animated,  lively. 

Animoso  (I) — Lively,  energetic. 

Anschwellen   Hi) — Crescendo, 

Antispastus  (L) — A  foot,  consisting  of 
two  long  between  two  short  sylla- 
bles. 

Antode    (Ok) — Responsive   singing. 

A  piaeere  (1)  —  (1)  At  pleasure;  not 
strictly  in  time,  ad  libitum;  (2) 
The  introduction  of  a  cadenza. 

A  poco  a  poco  (I) — More  and  more; 
by  degrees. 

A  poco  piu  lento  (I) — A  little  slower. 

A  poco  piu  ino8so   (I) — A  little  faster. 

Apotome  (Gk) — A  major  semitone, 
B  to  C. 

Appassionato    (I) — With    feeling. 

Appenato  (I) — With  an  expression  of 
suffering;  with  bitterness  or  grief. 

Appoggiatiira  (1) — A  note  leant  upon 
in  singing  or  playing,  as  a  grace 
note. 

A  piinto  (1) — In  exact  time,  precise. 


A   quatre  mains    (F),  A  quattro   mani 

(1) — For  four  hands  on  one  instru- 
ment. 

Arcato  (I) — With  the  bow,  as  opposed 
to  pizzicato,  plucked  with  the  finger. 

Aria   (1) — An   air,    tune,   song,    melody. 

bulTa      (1) — A     song     with     some 

degree  of  luimor  in  the  words,  or 
in  the  treatment  of  the  music. 

d'  entrata  (I) — The  first  or  en- 
trance air  sung  by  any  character  in 
an  opera. 

Armarius — Precentor. 

Arpe8:s:io   (I)     In  the  style  of  a  harp. 

Assai  (Ij — Very.  Allegro  assai,  very 
quick. 

A  tempo    (I) — In   time. 

A  tre  (I) — For  three  voices,  instru- 
ments, or  parts. 

Attacca  (I) — Commence  at  once,  with- 
out a  pause. 

A  vista  (I) — At  sight;  used  instead 
of  a  prima  vista,  at  first  sight. 


B. 


Barearola  (I),  Barcarolle  (F) — A  sim- 
ple melody  in  imitation  of  the  songs 
of  Venetian  gondoliers. 

Barre  (F) — In  guitar  or  lute  playing, 
the  pressing  of  the  forefinger  of  the 
left  hand  across  all  the  strings. 

Battere,  11  (I)— The  down-stroke  in 
beating  time. 

Bellicoso    (I) — Warlike,   martial. 

Ben  (I) — Well.  Ben  marcato,  well  and 
clearly  marked;  ben  sostenuto  or 
ben  tenuto,  well  sustained. 

Benedictus   (L) — Portion  of  a  Mass. 

Berceuse   (F) — A  cradle  song. 

Bindung:    (G) — Suspension. 

Bis   (L) — Twice. 

Bolero  (S)— A  Spanish  dance  in  triple 
measure,  accompanied  with  singing 
and  castanets. 


Bouche  fermee,  a  (F) — With  closed 
mouth;    humming. 

Bourree  (F) — A  dance-tune  in  com- 
mon time. 

Bravour  (G),  Bravura  (I)— Dash,  bril- 
liancy. 

Breit  (G)— Broadly. 

Brillante  (I  and  F)— Brilliant. 

BrIIlo   (I) — Joy,   gladness. 

Brio,  con   (I) — With  spirit,  vigor. 

Brioso  (I) — Joyfully,  vigorously. 

Brise    (F) — Broken   chords,  arpeggios. 

BuflTa,  feni..  Buffo,  mas.  (I) — Comic, 
Aria  Iniffa,  a  humorous  melody; 
opera   buffa,  a  comic  opera. 

Burletta  (I) — A  comic  operetta;  a 
farce  interspersed  with  songs. 


c. 


C  clef— The  clef  showing  the  position 
of  middle  C. 

Cabiscbol,  Cabiscola— The  precentor  in 
a  choir. 

Cachucha   (S)— A    Spanish   dance. 

Cacophony — Harsh-sounding    music 

Cadence — A  shake  or  trill,  run  or  divi- 
sion, introduced  as  an  ending,  or  as 
a  means  of  return  to  the  first  sub- 
ject. 

Cadenz    (G) — Cadence. 

Cadenza  (I)— (1)  A  passage  intro- 
duced toward  the  close  of  the  first 
or  last  movement  of  a  concerto;  (2) 
A  running  passage  at  the  end  of  a 
Tocal  piece. 

Calando  (I)— With  decreasing  Tolume 
of  tone  and  slackening  pace. 

Calata  (I) — Italian  dance  in  2-4  time. 

Calcando  (I)— Hurrying. 


Calore,  con   (I) — With  heat,  warmth. 

Caloroso  (I) — Warmly,  full  of  pas- 
sionate feeling. 

Canon— A  composition  in  which  the 
music  sung  by  one  part  is,  after  a 
short  rest,  sung  by  another  part 
note  for  note. 

Cantabile   (I) — In  a  singing  style.. 

Cantante   (I)— A  singer. 

Cantata  (I) — Originally  a  mixture  of 
recitative  and  melody  for  a  single 
voice,  but  now  a  short  work  In  the 
form  of  an  oratorio. 

Canticle — A  song  or  hymn  in  honor  of 
God,  or  of  some  special  sacred  event. 

Canto  (I) — The  upper  voice-part  in 
concerted  music. 

Cantor— Precentor. 


222 


MUSICOLOGY. 


Canzona,     Canzone     (I) — (1)     An     old 

form  of  soug;  (2)  An  iustrunieutiil 
composition  in  two,  three  or  four 
parts,  containing  contrapuntal  de- 
vices. 

Canzonetta  (I) — A  little  short  song, 
tune,  cantata,  or  sonata. 

Capischol    (L) — Precentor. 

Cappella,  alia  (I) — In  the  ecclesiastical 
style;    in  duple  time. 

Capo    (I) — Head,   commencement. 

Capricoio  (I) — A  freak,  whim,  fancy. 

Caprlocioso    (II — Whimsical. 

Caprice  (F) — Capricclo. 

Castrato  (I) — A  male  singer  with  a 
soprano  voice. 

Cavatina  (I) — A  melody  of  a  more 
simple  form  than  the  aria. 

Celere   (I) —  Quicls,  swift. 

Chaconne  (F)— A  slow  dance  In  % 
time. 

Chanson  (F)— (1)  A  song;  (2)  A 
national   melody;    (3)  A  part-soug. 

Chansonnette   (F) — A  little  song. 

Chef-d'oeuvre  (F) — The  master-work 
of  any  composer. 

Chlave  (I)— Key  or  clef. 

Chica — A  popular  dance  among  the 
Spaniards  and  the  South  American 
settlers  descended  from  them. 

Chromatic— Notes  not  belonging  to  a 
diatonic  scale.  A  chromatic  scale 
consists  of  a  succession  of  semi- 
tones. 

CIna  (F),  Cinque  (I)— A  fifth  part  In 
concerted  music. 


Coda  (I)  — (1)  The  tail  of  a  note;  (2) 
An  adjunct  to  the  ordinary  close  ol 
a  piece  or  song. 

Col,  Coir,  Colla,  CoIIo  (1 1— With  the. 

Coloratura  ( 1 )  — Divisions,  runs,  trills, 
cadenzas,  and  other  florid  passages 
in  vocal  music. 

Comodo   (I) — Easily,  without  haste. 

Con   (D— With. 

Contra  (I) — Against.  In  compound 
words  this  signifies  an  octave  be- 
low, as  Contra-gamba,  a  16-ft. 
ganiba. 

Contrappunto    (I) — Counterpoint. 

Contrapunkt    (fJ) — Counterpoint. 

Contrapuntal — Belonging  to  counter- 
point. 

Contra  tempo  (I)— Against  time.  (1) 
The  part  progressing  slowly  while 
another  is  moving  rapidly;  (2)  Syn- 
copation. 

Contrepartie  (F)— Counterpart,  oppo- 
site. The  entry  of  a  second  voice 
with  a  different  melody. 

Contretemps    (F) — .\gainst    time. 

Coranto  (U — An  Italian  form  of  the 
country  dance;    a   running  dance. 

Counterpoint — "The  art  of  adding  one 
or  moi-e  parts  to  a  given  melody." 

Couplet — (1)  A  verse  of  a  song;  (2) 
Two  notes  occupying  the  time  of 
three. 

Crescendo  (I) — Increasing;  a  grad- 
ual increase  in  the  force  of  sound. 

Crotchet — A  quarter  note,  one-fourth 
of  the  value  of  a  semibreve. 


Da  (I)— From,  by,  of,  for,  etc. 

Da  capo,   or   D.  C.    (I) — From    the   be- 


Da  capo  al  fine   (I) — From   the  begin- 
ning to  the  sign  Fine. 

al    segno     (1)  — Repeat    from    the 

beginning  to  the  sign  S. 

al  segmo  (I)— From  the  sign  S. 

ecrescendo      (I)— Decreasing     gradu 


;illv  the  volume  of  tone. 

ecupIet^.V  group  of  ten  notes  played 

111    tVio    fim<^    rif   aitrlit-    or  .f(inr 


pcuplet — .\  group  or  ren  note 
in  the  time  of  eight  or  four. 
Diatonic— One  of  the  three  genera  of 
music  among  the  Greeks,  the  other 
two  being  the  chromatic  and  en- 
harmonic. 


Diminuendo  (I)— Decreasing  in  power 
of  sound. 

Ditone — An  interval  of  two  major 
tones. 

Divertimento  (I) — (1)  An  instrumen- 
tal composition  of  a  light,  pleasing 
character;    (2)   Pot-pourri. 

Dolce  (I)— Softly,  sweetly. 

Dolente,  Doloroso  (H  In  a  plaintive, 
sorrowful   style;    with   sadness. 

Dominant— The  fifth  degree  of  a  scale. 

Droite  fF)— Right;  as  main  drolte, 
the  right  hand. 

Duple  time — Even  time. 


E. 


Echelle  (F)— A  so.ile;  as,  ^chelle  chro- 
matique,  chromatic  scale;  echelle 
diatonique,  diatonic  scale. 

Eclogrue  (F)— A  shepherd's  song;  a 
pastoral  piece. 

Rlferig:   (O)— Zealously,  ardently. 

Kinfalt  (O)— With  simplicity  and  dig- 
nity. 

Einlgrem  Pomp,  mit  (O)— In  a  some- 
what pompous  manner. 

Einsclilafen  (O)— To  slacken  pace  and 
dimlnlsli  the  power. 


Kin  wenlg  lebendigr  (01— R.ither  lively. 

Blegant  (F).  Elegante  (1)— With  ele- 
gance of  style. 

Emozione,  con    (I) — With   emotion. 

Empflndung    (O)— Emotion,    passion. 

En  badinant    (F)— Scherzando. 

Energia,   Energico    (1) — With   energy. 

Enfler  (F) — To  swell;  to  increase  in 
sound. 

Ensemlile  (F)— Together ;  the  whole. 
The  general  effect  of  a  musical  per- 
formance. 


MUSICAL   DICTIONARY 


223 


Entr'acte  (F) — Music  played  between 
the  acts  of  au  opera,  drama,  etc. 

Kntrante  (I),  JJntree  (F)— Eutry,  in- 
troduction, or  prelude. 

KntuMiasmo   (I) — Witli  enthusiasm. 

Kntwurf   (G) — A  sketch. 

ICpiceclion   ((Jli) — A  dirge;    elegy. 

Kpilenia   ((ik) — Vintage  songs. 

Kpithalamium  (Gk) — A  nuptial  song. 

Epoile   (Gk) — An  after-song. 


Krhaben   (G) — Exalted,  sublime. 
Krn»itlicli    (G) — Earnestly,   fervently. 
Krotique  (F) — A  love-song. 
K8i>agnuolo  (I)— In  the  Spanish  style. 
Espressivo  (I) — Expressive. 
E»<tinKuen)lo    (I)— Dying    away,    grad- 
ually reducing  both  power  and  pace. 
Etude  (F)— A  study,  exercise,  lesson. 
Eveille  (F)— Sprightly,  quick,  lively. 


F  clef— The  bass  clef. 

Facile  (F)— Easy. 

Falsetto  (1) — The  artificial  or  supple- 
menting tones  of  the  voice,  higher 
than  the  chest  or  natural  voice. 

Fandangro  (S) — A  lively  Spanish  dance 
in  triple  time. 

Fantasia  (I),  Fantalsie  (F),  Fantasie 
(G) — A  composition  in  which  form  is 
subservient  to  fancy. 

Farandola  (I),  Farandoule  (F) — An  ex- 
citing dance,  popular  among  the 
peasants  in  the  south  of  France  and 
the  neighboring  part  of  Italy. 

Fausset  (F)— Falsetto. 

Feldmusili  (G) — Military  music. 

Fermamente  (I) — Firmly,  with  deci- 
sion. 

Ferniata  (I) — A  pause. 

Feroce  (I) — Wildly,  fiercely. 

Fertlg  (G) — Quick,  dexterous. 

Fervore,  con  (I) — With  fervor. 

Fest  (G)— (1)  A  festival;    (2)  Firm. 

Festivo  (1) — Festive,  solemn. 

Festoso  (I) — Joyous,  gay. 

Feuer  (G) — Fire,  ardor,  warmth. 

Fiacco  (I) — Weak,  weary,  faint. 

Fieramente  (I) — Proudly,  fiercely. 

Figured  bass — A  bass  having  the  ac- 
companying chords  suggested  by 
certain  numbers. 


Filer  le  son  (F) — To  prolong  a  sound. 

Fin   (F)— The  end. 

Finale  (1) — The  last  movement  of  a 
concerted  piece,  sonata,  or  sym- 
phony; the  last  piece  of  an  act  of 
an  opera,  or  in  a  program. 

Fine  (I)— The  end. 

Flebile  (I) — In  a  mournful  manner. 

Fliessend   (G) — Fluently,  softly. 

Flueclitlgr  (G) — Light,  rapid. 

Foco  (I) — Fire,  spirit. 

Fort  (F),  Forte  (I)— Loud. 

Fortissimo  (I) — Very  loud. 

Fortississimo  (I) — As  loud  as  possible. 

Forza,  con  (I) — With  emphasis. 

Forzando  (I) — Forcing.  Emphasis  or 
musical  accent  upon  specified  notes 
or  passages. 

Francaise  (F) — A  dance  in  a  triple 
measure. 

Pretta,  con  (I) — With  speed,  haste. 

Friscli  (G)— Lively. 

Froehlich  (G) — Joyous,  cheerful,  gay. 

Fugre  (G) — A  fugue. 

Fugue — A  polyphonic  composition  con- 
structed on  one  or  more  short  sub- 
jects or  themes. 

Funebre   (F) — Funereal,   mournful. 

Fuocp,  con  (I) — With  fire,  energy. 


G. 


G  clef— The  treble  clef. 

Gal  (F),  Gajo  (I) — Lively,  merry,  gay. 

Gamut — The  scale. 

Gauche  (F)— Left. 

Gavotte  (F),  Gavotta  (I)— A  dance- 
tune  of  a  lively  yet  dignified  char- 
acter. 

Gedackt  (G) — Covered  or  closed. 

Gefuehl,  mit  (G)— With  feeling,  ex- 
pression. 

Geitneipt  (G) — Pizzicato. 

Gesang  (G) — Singing,  song,  cantata, 
hymn,  etc. 

Geschwind  (G) — Quick,  rapid. 

Gestofssen   (G) — Staccato. 

Getragen  (G) — Sustained. 

Gioco,  con   (I) — Sportively,  playfully. 

Giusto  (I) — Strict,  correct,  moderate. 

Glaenzend    (G)^Brillinnt. 

Glissando  (I)  — (1)  Sliding  the  tips  of 
the  fingers  along  the  keys  of  the 
piano;  (2)  A  rapid  slur  in  violin- 
playing. 


G^Iockenspiel  (G)  —  (1)  .\n  instrument 
made  of  bells  tuned  diatonically  and 
struck  with  hammers;  (2)  An  organ 
stop  of  two  ranks. 

Graduate  (L) — A  piece  of  music  per- 
formed between  the  reading  of  the 
Epistle  and  Gospel  in  the  Roman 
Church. 

Grandloso  (I) — In  a  lofty  manner. 

Grave  (L..  I.,  F.,  E.)  — (1)  Deep  in 
pit<-h ;  (2)  Slow  in  pace,  solemnly, 

Gravemente  (I) — With  dignity,  grav- 
ity, earnestness. 

Grazioso  (I) — Gracefully,  elegantly. 

Gregorian — Plain-song. 

Gruppetto  (1) — Notes  grouped  as  a 
cadenza,  division,  or  ornament. 

Guaraclia  (S) — A  lively  Spanish  dance. 

Gusto,  con  (I) — With  taste  and  ex- 
pression. 


224 


MUSICOLOGY 


H. 


Hailing— A  Norwegiau  dauce. 
UochzeitmarNch   (G) — Weddiug  niaiili. 
Uosanna — (1)    An   exclatuatiou,    "Save, 


I    pray,"   formed   from   Ps.   118:   25; 
(2)  I'art  of  the  Sauetus  in  the  Mass. 
Ilurtig  (<i) — Nimble,  quick,  agile. 


Iambic — Having  a  short  and  a  long 
syllable  alternately. 

II  fine  (I)— The  end. 

Imperioso  (I) — With  grandeur,  dig- 
nity, imperiou.sly. 

Impeto,  con  (I) — Impetuously. 

Impresario  (I) — A  designer,  conductor 
or  manager  of  a  concert  or  opera 
party. 

Impromptu  (I)  —  (1)  Music  written  or 
played  without  previous  prepara- 
tion; 12)  .\  piece  in  the  style  of  an 
improvisation. 

In  alt  (I) — All  notes  in  the  first  octave 
beyond  the  range  of  the  treble  stave. 


In  altissimo  (I)— All  notes  beyond  the 
range  of  the  first  octave  in  alt. 

Intermezzo   (I) — An  interlude. 

Intonation — (1)  The  method  of  pro- 
ducing sound  from  a  voii-e  or  an 
instrument;  (2)  Singing  or  playing 
in  perfect  tune;  (3)  The  method  of 
chanting  certain  portions  of  the 
Church  services. 

Intrada  (I» — .\n  interlude  or  entr'acte. 

Introit — An  iintiplion  sung  while  the 
priest  proceeds  to  the  altar  to  cele- 
brate Mass. 

Ite  mifssa  est  (L) — The  concluding 
words  of  the  Mass  in  the  Romish 
Church. 


J. 


Jaleo — A  national  dance  of  Spain. 
Jodie— A    peculiar    method    of   singing 


adopted  by  the  Swiss  and  Tyrolese. 
Jota — A  Spanish  dance. 


K. 


Kalamaika— A  Hungarian  dance. 

Kapelle  (G) — A  word  formerly  applied 
to  a  private  band,  but  now  used  to 
denote  any  band. 

Kapellmeister  (G)— The  leader  or  con- 
ductor of  a  band  of  music. 

Kircbe  (G)— Church. 


Klagrend  (G) — Mournfully,  plaintirely. 
Kraeftig  (G) — Energetically. 
Kreol  (D) — A  dance  similar  to  the  reel. 
Kreuz  (G) — The  sign  for  a  sharp. 
Kriegslied  (G) — A  battle-song. 
Kunstpfeifer  (G) — Town  musician. 


L. — The  letter  employed  as  the  ablire- 
viation  of  the  word  left  or  linke  (<i). 

L,a  destra  (D— The  right  hand. 

Lacrimoso  ( I )— Mournfully. 

Lai  (F»— A  lay.  song. 

Largamente  (I) — Slowly,  widely,  freely. 

Larghetto  (I)— .\t  a  slow  pace,  but  not 
so  slow  as  largo. 

Larghissimo  (I) — Exceedingly  slow. 

Largo  (I)— Slow,  broadly. 

Laut  (G) — Loud,  forte;    sound. 

Lebendig  (G) — Lively. 

Lebhaft    (G)     Lively. 

Lebbaftesten'    (G) — With  extreme  live- 
liness. 

Legando  (I)— Tied,  connected. 

Legatissimo  (I)— Exceedingly  smooth. 

Legato  (I) — Round,  close,  connected. 

Legatura  (I) — A  bind,  brace,  or  tie. 

Leggiere  (I)— Very  lightly,  rapidly. 

Leggiermente  (I)  — With  lightness. 

Leicbt   (Gl-  Easy,  light. 

Leidenscliaftlicb    ((i)     Passionately. 

Leise  ((',)    -Softly,  lightly. 

Leitmotiv  (G) -.\  theme  constantly  re 
curring  in  association  with  a   parti- 


cular person  or  action  throughout 
an  opera. 

Lent  (F)— Slow. 

Lentando  (I) — Becoming  slower  by  de- 
grees ;    slackening  the  time. 

Lento  (D- Slow. 

Lesto  (I) — Light,  lively,  cheerful,  gay. 

Libretto — The  book  containing  the 
words  of  an  oratorio,  opera,  etc. 

Lied  (G) — A  composition  of  a  simple 
cliaracter,  which  is  complete  in  itself; 
,'1  song. 

Tyje<lerbiicb   (G)  —  Song- book. 

Liederspiel  (<!) — .V  play  with  songs  of  a 
I>oi)ular  cliaracter  introduced  iiito  it. 

Liedertafel  ( C  I-  Literally,  Song-table. 
A  society  meeting  tor  the  practice  of 
part-songs  for  men's  voices. 

Linke  Hand  (<;)— Left  hand. 

Lire  (K) — A  lyre  or  harp. 

Loco  (1) — In  its  proper  place.  A  direc- 
tion to  return  to  the  proper  pitch 
after  having  played  .•in  oct:ive  higher. 

Loure  iir  liOiivre   (I'")     .\  dtiiice. 

Liistig  ((;>    Merry,  merrily. 

Lyre — One  of  the  most  ancient  stringed 
instruments. 


MUSICAL   DICTIONARY 


225 


M. 


M — Abbreviation  of  mezzo,  mano,  main, 
manual. 

Ma(lrig:al — A  word  of  doubtful  orijiin. 
It  became  a  general  term  for  secular 
compositions,  of  which  there  were 
various  forms  differing  in  style. 

Maestoso  (I) — With  dignity,  majesty. 

Maestro  (I) — Master. 

Maggriolata  (I) — A  song  sung  iu  cele- 
bration of  the  month  of  May. 

Main  (F)— The  hand. 

3Iaitre  de  chapelle  (F) — Choirmaster. 

Malinconico  (I) — With  sadness,  sorrow. 

Maniere  (F) — As  maniera  affettata,  an 
affected  style;  maniera  languida,  a 
languid,  lifeless  style. 

Maennerstimnien   (O) — Men's  voices. 

Mano  (I)— Hand. 

Marzlale  (I) — In  a  martial  style. 

Mazurka — A  Polish  dance  of  lively, 
grotesque  character 

Meistersinger  (G) — A  title  given  to  the 
most  renowned  musician  of  a  town- 
ship or  district  in  Germany  during 
the  Middle  Ages. 

31elangre  (F) — A  medley. 

Meno  (I) — Less. 

Mestoso  (I) — Sad,  pensive. 

Mezzo  (I) — Half  or  medium. 

Minuet — The  name  of  a  graceful  dance 
iu  triple  time. 


Miserere — The  51st  Psalm  sung  in  the 
Tenebrae  service  in  the  Roman  Cath- 
olic Church. 

Mit  (G I— With. 

Moderate  (I> — Moderately. 

Moll  (G)— Minor. 

Molto  ID— Much,  very. 

Monotone,  to — To  recite  words  on  a 
single  note  without  inflections. 

Morceau  (F) — .\  piece;  a  small  compo- 
sition of  an  unpretending  character. 

Mordente  (I)— A  beat,  or  turn,  or 
passing  shake. 

Morgenlied  (G) — Morning  song. 

Mosso  (I) — Moved. 

Motet — .\  vocal  composition  in  har- 
mony, now  generally  set  to  sacred 
words. 

Motiv  (G)  — (1)  The  sort  of  movement 
indicated  by  the  opening  notes  of  a 
sentence;  (2)  A  subject  proposed  for 
development. 

Munter   (G) — Lively. 

Musette  (F)  — (1)  A  small  bagpipe;  (2) 
A  melody  of  a  soft,  sweet  character, 
written  in  imitation  of  the  bagpipe- 
tunes;  (.3)  Dance-tunes  and  dances 
in  the  measure  of  those  melodies; 
(4)  A  reed-stop  on  the  organ. 

Muthig  (G)— With  spirit. 


N. 


Nacli  Belieben  (G) — Ad  libitum. 

Naehspiel   (G) — A  postlnde. 

Nachtstueeke  (G) — Night-visions.  The 
name  of  four  pianoforte  pieces  by  R. 
Schumann. 

Naeh  und  naeh  (G) — By  little  and  lit- 
tle, by  degrees. 

Naivement  (F) — Artlessly,  unaffectedly. 

Nobilmente  (I)— With  grandeur,  nobly. 


Noel  (F) — "Good  news."  A  word  used 
as  a  burden  to  carols  at  Christmas. 

Non  (I)— Not. 

Notturno  (I) — Originally,  a  kind  of 
serenade;  now  a  piece  of  music  of 
a  gentle  and  quiet  character. 

Nuances  (F) — Shades  of  musical  ex- 
pression. 


Obbligato  (I) — An  instrumental  part 
or  accompaniment  of  such  impor- 
tance that  it  cannot  be  dispensed 
with. 

Offertoire  (F),  Offertory— A  piece  of 
music  performed  during  the  offer- 
tory. 

Opera    (I)— A    dramatic  entertainment 


of  Italian  origin  in  which  music 
forms  an  essential  part. 

Opus   (L) — A  work. 

Oratorio  (I) — A  composition  for  voices 
and  instruments  illustrating  some 
sacred  subject. 

Overture — An  instrumental  piece  writ- 
ten as  a  prelude  to  an  opera,  ora- 
torio, or  other  work. 


Paean  (Gk» — The  ancient  choral  song 
addressed  to  Apollo.  Sung  before  or 
after  a  battle. 

Part-sonff-  A  vocal  composition,  hav- 
ing a  striking  melody  harmonized 
by  other  parts 

Passionato  (I)  — In  an  impassioned 
manner. 


Pastoral,  Pastorale  fl)  —  CI)  A  simple 
melody  in  6-8  time  in  a  rustic  style; 
(2)  A  cantata,  the  words  of  which 
are  founded  on  [lastoral  incidents; 
(.'?)  A  complete  symphony,  wherein 
a  series  of  pastoral  scenes  is  de- 
picted by  siiuiid-painting,  without 
the  aid  of  words. 


226 


MUSICOLOGY 


Pafetiro  III,  Fathetique  (Fl  — (1)  Pa- 
tbetic;    (J)   In  a  pathetic  iiiauiier. 

I'entatonir  h«-ale  -The  iiaiiie  jiiven  to 
the  ancient  musical  scale,  which  is 
easiest  described  as  that  formed  by 
the  black  keys  of  the  piano-fortp. 

I'etit  choeur  (F) — The  chorus  which 
originally  consisted  of  three  parts 
only. 

Peu  a  peu  (F)— Little  by  little. 

Phrasing: — The  proper  rendering  of 
music  with  reference  to  its  melodic 
form. 

Piacere,  a  (11— At  pleasure. 

Pianissimo  (I) — Extremely  soft. 

Pianississimo  (I) — As  softly  as  possi- 
ble. 

Piano  (D— Softly. 

Piu  (I)— More. 

Pizzicato  (I) — A  direction  to  violinists 
to  produce  the  tone  by  plucking  the 
string  with  the  finger. 

PlagHl  cadence — The  cadence  formed 
when  a  subdominant  chord  imme- 
diately precedes  the  final  tonic  chord. 

Poco  (I»— A  little. 

Poi  (I)— Then. 

Poiaoca  (I) — Polish.  A  title  applied 
to  melodies  written  in  imitation  of 
Polish  dance-tunes. 


Polka  A  Rohemian  dance  of  world- 
wide popularity,  in  l.'-4  time. 

I'omx»oso  (1 )  —  Pomi)ously. 

Portamento  (1) — A  lifting  of  the  voice, 
or  gliding  from  one  note  to  another. 

Postlude — A  concluding  voluntary;  a 
piece  played  at  the  end  of  service. 

Pot-pourri  (F)— A  medley;  a  collec- 
tion of  various  tunes  linked  together. 

Pralltriller   (G) — A  transient  shake. 

Precentor — An  official  in  a  cathedral 
who  leads  and  directs  the  choir,  etc. 

Precipitato  (I) — With  precipitation. 

Preciso  (I)— With  exactitude. 

Prelude — A  movement  played  before, 
or  an  introduction  to,  a  musical 
work  or  performance. 

Premiere  (F) — First. 

Prestissimo  (I) — Very  fast  indeed. 

Presto   (I)— Fast. 

Prima  (I,  fern.) — First. 

Prime  (I,  masc.) — First. 

Proscenium— (1)  The  quadrangular 
space  behind  the  logeum  or  stage; 
(2)  The  stage  front. 

Provencales  —  Troubadours  of  Pro- 
vence. 

Psalter— A  book  of  Psalms. 


Q. 


Quadrille — A    well-known    dance,    con- 
sisting of  five  movements. 


Quasi 

of. 


(I)— As    if,    or    in    the    style 


Raddolcendo  (I)— With  gradual  soft- 
ness and  sweetness. 

Rallentamento  (D— At  a  slower  pace. 

Rallentando  (1)— Getting  gradually 
slower. 

Rapidita,  con   (I)— With   rapidity. 

Recitative — Musical  declamation;  a 
kind  of  half-speaking  and  half-sing- 
ing; a  composition  without  any  de- 
cided or  rhythmical  melody. 

Reel  (old  E)—.\- lively  rustic  dance, 
popularly  supposed  to  be  Scotch,  hut 
prol)ably  of  Scandinavian  origin. 

Reveil  (old  E)-  Music  which  wakens 
from  sleep.  .\  signal  given  by  drum 
to  soldiers  at  d.iwn. 

KImpsodie  (G).  Rhapsody— A  compo- 
sition of  irregular  form,  and  in  the 
style  of  an  improvisitioii. 

Rh.vthm— The  arrangement  of  niusicil 
phrases  or  sentences  in  regular  met- 
rical form,  as  regards  accent  ■■md 
nuantity. 

Rigore   (I)     Strictness. 

Risoluto   (1)     With  resolution. 

Kitardando  (I)-  With  gradually  in- 
creasing slowness  of  pace. 


Ritennto  (I) — Holding  back  the  pace. 

Robusto  (I) — Robust,  strong,  powerful. 

Role  (F)  The  part  in  a  drama  as- 
signed to  an  iictor. 

Romance  (F.  SI,  Romanza  (11 — Any 
simple  rhytlimical  melod.v  which  is 
suggestive  of  a  romance. 

Romanesca  (I) — .\n  Italian  dance. 

Komcra — .\   Turkish  dance. 

Rondeau  (F».  Rondo  (I)  A  composi- 
tion goner.ally  in  two  parts,  with  the 
principal  subject  often  repe.'ited. 

Roulade  (F) — An  embellishment;  a 
flourish. 

Round — .\  composition  in  which  sev- 
eral voices,  starting  at  stated  dis- 
tances of  time  from  eacli  ottier.  sing 
each  the  s.ime  music,  the  combina- 
tion of  all  the  parts  producing  cor- 
rect liarmony. 

Roundel  A  rustic  song;  a  dance  in 
wliicli  all  .ioin  hands  in  a  ring. 

Roundela.v — i\)  A  poem.  cert.\in  lines 
of  wliich  are  repeated  at  intervals; 
(2)  The  tune  to  which  a  roundelay 
was  sung. 


MUSICAL  DICTIONARY 


227 


Salto  (I) — (1)  A  dance  in  which  there 
is  much  leaping  and  skipping;  (2) 
A  leap  or  skip  from  one  note  to 
another  beyond  the  octave;  (.'{) 
Counterpoint  in  which  the  part 
added  moves  in  skips. 

Sans    (F)— Without. 

Saraband,  Sarabanda  (I),  Sarabande 
(F) — A  Spanish  dance  of  Moorisli 
origin,  for  a  single  performer,  ac- 
companied with  castanets. 

Satz  (G) — A  theme,  subject,  composi- 
tion, piece,  movement. 

Srherzando  (I)  —  (1)  Playful,  lively: 
(2»  A  movement  of  a  droll  character. 

Scherzo  (I) — Literally,  a  jest,  applied 
to  a  movement  in  a  sonata  or  sym- 
phony, of  a   sportive  character. 

Srhlummerlied    (G) — A   slumber-song. 

Srhluss   (G) — The  conclusion;    finale. 

.Srhlu!«sstueck  (G) — Finale. 

Schmelzend  (G) — Literally,  Melting 
away.     Dying  away ;    diminishing. 

Srhnell    (G)— Quick. 

Srhottisehe  (G)— Literally,  The  Scotch 
dance.  A  slow  dance  of  modern  in- 
troduction, in  2-4  time. 

Score — A  copy  of  a  musical  work  in 
whicli  all  the  component  parts  are 
shown  either  fully  or  in  a  com- 
pressed form. 

Sdegno,  con  (I) — Scornfullv  ;  disdain- 
fully. 

Sec  (F) — Dry,  unadorned,  plain. 

Seoondo   (I) — Second. 

Segno  (I)— The  sign  S. 

Segue   (I) — Follows,  succeeds. 

Seguidilla  (S) — A  lively  Spanish  dance, 
similar  to  the  country-dance. 

Semplice   (I) — Pure,  plain,  simple. 

Senipre  (I) — Always,  ever. 

Senza   (I)— Without. 

Sequence — The  recurrence  of  a  har- 
monic progression  or  melodic  figure 
at  a  different  pitch  or  in  a  different 
key  to  that  in  which  it  was  first 
given. 

Serenade — (1)  Originally  a  composi- 
tion for  use  in  the  open  air  at  niglit. 
generally  of  a  quiet,  soothing  char- 
acter :  (2)  A  work  of  large  propor- 
tions in  the  form  of  a  symphony. 

Sestetto  (1) — A  composition  for  six 
voices  or  instruments. 

Sforzando  (I) — Forced. 

Siciliana  (I) — A  graceful  dance  of  the 
Siiilian  peasantry. 

Siegeslied  (G) — A  song  of  triumph. 

.Silhouettes  (F) — Sketches;  recollec- 
tions. 

Singspiel  (G) — Opera. 


Slargando   (I) — Widening,  opening. 

Slentando  (I) — Slackening  the  time. 

Soave   (I) — Agreeably,  delicately. 

Soggetto   (1) — Subject,  theme,  motive. 

Solennel,  le  (F) — Solemn. 

Sol-faing — A  vocal  exercise  in  which 
the  notes  are  called  by  the  several 
names  Do,  Re,  Mi,  Fa,  Sol,  La.  Ti. 

Sonata — A  composition  consisting  of 
three  or  four  movements,  generally 
for  a  solo  instrument,  and  in  sym- 
phonic form. 

Sonatina  (I),  Sonatine  (F) — A  short 
sonata. 

Sopra  (I) — Above,  before,  over,  upon. 

Sorda,  Sordo  (I) — MuflJed  with  a  mute. 

Sortie  (F) — A  voluntary  played  at  the 
close  of  a  service. 

Sostenuto   (I) — Sustaining. 

Sotto  (I) — Below,  under. 

Sous  (F) — Under. 

Spianato   (I) — Smooth,  level,  even. 

Spirito,  con  (I) — In  a  spirited,  lively, 
animated,  brisk  manner. 

Staccato  (I)— Detached,  taken  off. 

Stanza  (I) — A  verse  or  subdivision  of 
a  poem  ;    a  strophe. 

Stentato  (I) — Forced,  emphasized. 

Stimme  (G)  — (1)  The  voice;  (2)  Sound; 
(.'5)  The  sound-post  of  a  violin  or 
violoncello;  (4)  A  part  in  vocal  or 
instrumental  music;  (.5)  An  organ- 
stop  or  rank  of  pipes. 

Strepitoso  (I) — Noisy,  impetuous. 

Stringendo  (I) — I'ressing,  hastening  on 
the  time. 

Stueck  (G) — A  piece,  air,  composition. 

Suave  (I) — Sweet,  agreeable,  pleasant. 

Sub  (D— Under. 

Sub-bass — A  pedal  register  in  the 
organ,  of  Z2-tt.  tone. 

Subdominant — Tlie  fifth  below  or  the 
fourth  above  any  key-note. 

Suite  (F) — A  set,  series,  or  succession 
of  movements  in  music. 

Super  (L) — Above,  over. 

Suess  (G)— Sweet. 

Symphony — (1)  A  composition  for  an 
orchestra,  similar  in  construction  to 
the  sonata,  which  is  usually  for  a 
single  instrument;  (2)  Formerly 
overtures  were  called  symphonies; 
(.'?)  The  introductory,  intermediate 
and  concluding  instrumental  parts 
of  a  song  or  other  vocal  piece  are 
also  called  symphonies. 

Syncopation — Suspension  or  alteration 
of  rhythm  by  driving  the  accent  to 
tliat  part  of  a  bar  not  usually 
accented. 


T. 


Takt  (G) — Time,  measure,  bar. 

Tanto  (1) — So  much;    as  much. 

Tarantella     (I) — A     rapid     Neapolitan 
dance  in   triplets,  so  called   because  I 


it  was  thought  to  be  a  remedy 
against  the  supposed  poisonous  bite 
of  the  tarantula  spider. 

Tardamente  (I) — Slowly. 

Tardo  (I)— Slow,  dragging. 


228 


MUSICOLOGY 


Tedinik  (G) — A  general  name  for  the 
systems,  devices  and  resources  of 
musical  art. 

Tenia  (I) — A  theme  or  subject;  a 
melody. 

Tenii)<>  (II~Tiiiie  or  measure. 

Temps  (I'')— Time;  the  parts  or  divi- 
sions of  a  l)ar. 

Tenu  (I''),  Teniito  (I)— Held  on;  sus- 
tained;   kept  down  for  the  full  time. 

Terzetto  (1) — A  composition  for  tliree 
lierformers. 

Tetraehord — A  scale-series  of  four 
notes.  The  word  in  its  modern  sense 
sijrnifies  a  half  of  tlie  octave  scale. 

Thesis  (Gk)— (1)  In  metre,  the  heavy 
tone  or  vocal  accent;  (2)  In  rliythm, 
tlie  non-accent,  or  up-beat. 

Threnody  (Gk) — An  elegy,  or  funeral 
song. 

Tie— (1)  A  curved  line  placed  over  two 
or  more  notes  in  the  same  position 
on  the  stave,  to  show  they  are  to  be 
played  as  one;  (2)  When  two  or 
more  quavers,  semiquavers,  etc.,  are 
united,  instead  of  being  written  with 
separate  tails,  they  are  said  to  be 
tied. 


Timbre  (F) — Quality  of  tone  or  sound. 

Timoroso  (I) — Timorous;  with  liesi- 
t.ition. 

Toccata  (I)  —  (1)  A  prelude  or  over- 
ture; (2)  Compositions  written  as 
e.vercises;  ('.'<)  A  fantasia;  (4)  A 
suite. 

Tonfiiehrungr  (G)  —  (1)  A  melodic  suc- 
cession;    (2(   Modul.ition. 

Tonic, Tonico  (1  I,  Tonique  (F)  — (1)  The 
key-note  of  .my  scale;  tlie  ground- 
tone  or  basis  of  a  scale  or  key;  (2i 
The  key-chord  in  which  a  piece  is 
written. 

Tonltuenstler  (G) — A  musir'ian  ;  a  niu- 
sii-al  artist. 

Tremolo   (I)— Trembling. 

Tranquillo    (I) — Tranquilly,  calmly. 

Tres   (F) — Very. 

Troppo  (I) — Too  much. 

Tufti  (I» — All.  Every  performer  to 
take  part  in  the  execution  o-f  the 
passage  or  movement. 

Tyrolienne — (1)  A  song  accompanied 
with  dancing;  (2)  Popular  songs  or 
melodies  in  which  the  jodle  (q.  v.) 
is  freely  used. 


u. 


T'ebellilang 

phony. 


(G)  —  Discord,      caco- 


Un  (I)— One. 

In  poco  (I)— A  little. 


V. 


Va  (I)— Go  on. 

Vaudeville  (F)— A  play  with  songs 
set  to  popular  tunes. 

Veloce  (I)— Rapid,  swift. 

Versette  (G)— Short  pieces  for  the 
organ  intended  as  preludes  or  vol- 
untaries. 

Verte  (D— Turn  over. 

Vibrante  (I) — Vibrating,  tremulous. 

Viel  (G)— Much. 

VierBesanfi:  (G)— Song  for  four  parts. 

VierliaendiK   ((J)— For  four  hands. 

Vif  (F>     Tiively,  brisk,  quick. 

Vigoroso   (H— Vigorous,  bold,  forcible. 

Vistamente  (I)— Briskly,  quickly. 

Vivace  (1)— Lively,  quickly,  sprightly. 


Vivarissimo  (I) — Very  lively. 

Vivamente  (I),  Vivement  (F) — Lively. 

Voce  (1).  Voix  (F),  Vox  (L)— The 
voice. 

Volante  (I)  —  Flying.  .\pplicd  to  the 
execution  o-f  a  rapid  series  of  notes, 
either  in  singing  or  pl.iying. 

Volkslied  (G) — A  popular  song. 

Voluntary — An  organ  solo  played  be- 
fore, during  or  after  any  office  of 
the  rinirch ;  lience.  called  respec- 
tively introductory,  middle  or  con- 
cluding voluntary. 

Vorsaenger  (G) — Precentor. 

Vorspiel   (G) — Prelude;    overture. 


w. 


Wecbselgesang      (G)— Responsive 

antiphonal  song. 
Wehmuth  (G)— Sadness,  sorrow. 


WieKPnlied    (G) — .\   lullaby;    a   cradle- 
song. 
Wuerde,  mit   (G))— Witli   dignity. 


Zaleo— A  Spanish  national  dance. 
/aertlirh  (G)— Softly,  delicately. 
Zeichen    (G)— A  musical  sign,  note,   or 
character. 


Zeloso  (I) — Zealous,  energetic. 
Ziemlirli   (Gl — Moderately. 
Zurueckhaltungr  (G)— Retardation. 


-lO — 1 


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COMBINKD    MAJOR    AND    MINOK    KEY    TABLE. 


I  I   I  ;  I   I  II" 

I  I  1  1  1  I  I  I  I  I  I  ** 

(See   pM;re  ;>7 :   L'O.  lil) 

ria<'C    (■()nil)ii)ati()ii     triad    foriimla     (cut  from   page  220)  on  the  ciirveil  lines 

between      the     major     and      minor     liey    tahlfs:     and     liold     in  place     liy  clip?! 
as  in  Chart  1    Csee  slit  marks). 


c    C    =" 

•5  a  (B 


r       5  S   c   g 


>>  >  5^  -S 


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01 


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:i'2  2  " 

>>  T3    2     O 


ti! 


3  OS 


.5- as 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

MUSIC  LIBRARY 

This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 

Renewed  books  are  subject  to  immediate  recall. 


DfO  2      iqRfl 

JAN    21963 

fFRl-^lSf^ 

OCT    9  1972 

DEC!  5 1974 

LD21A-10m-5,'65                            ,,    .  General  Library 
(F4308slO)476                              University  of  California 

Berkeley 

MT6.L88 
C0371 40828 

U  C.  BERKELEY  LIBRARIES 


CD37mDfiEa 


DATE  DUE 


Music  Library 

University  of  California  at 
Berkeley 


HS'-iiiiMiiiiiiiiiiiiiiiiiiiiii 


